# Computation of weight 1 modular forms with exotic representations

Kieran Child
University of Bristol<EMAIL_ADDRESS>University of Bristol
Beacon House
Queens Road
Bristol
BS8 1QU
UK

###### Abstract.

We present a deterministic algorithm for computing spaces of weight 1 modular
forms with exotic representations. This algorithm is an improved version of
Schaeffer’s Hecke stability method, utilising the author’s previous work on
the twist-minimal trace formula for weight 2 holomorphic forms, and presenting
a method of lifting forms from characteristic $p$.

The algorithm was used to compute all such forms with level at most 10,000.
Together with Sutherland’s computation of dihedral forms, this allows us to
present the dimensions of all weight 1 newform spaces up to level 10,000.

##  1\. Introduction

The Langlands program conjectures associations between cuspidal automorphic
forms of $\text{GL}_{n}$ and irreducible, $n$-dimensional representations of
the absolute Galois group
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Following significant
results in [13] and [17], this association has been proven for the
2-dimensional, odd case. We have a bijection:

$\left\\{\begin{matrix}\text{Weight 1 newforms of}\\\ \text{level $N$ and
character
$\chi$}\end{matrix}\right\\}\longleftrightarrow\left\\{\begin{matrix}\text{Odd,
irreducible, 2-dimensional}\\\ \text{representations of
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ with}\\\
\text{conductor $N$ and determinant character $\chi$}\end{matrix}\right\\}.$
(1)

Hecke eigenvalues of the newform are equal to traces of Frobenius elements in
the representation. Despite this bijection having been established, concrete
examples are difficult to construct. This is especially so when the projective
image of the representation is not dihedral.

Typical approaches to computing Fourier expansions of arbitrary newforms, such
as by modular symbols (see [12]) or trace formulae (see [10]) require that the
weight be at least 2. Dimension formulae, such as in [22], also have this
restriction. Instead, a number of specialised methods for computing weight 1
forms have been proposed. See [11, Section 13.6.1] for a review of three such
methods. The most efficient in general is the Hecke stability method,
presented in [23].

While the computation of weight 1 newforms of a given level and character is
slow, there are two related spaces which can be computed much quicker.
Firstly, weight 1 newforms associated with representations with dihedral
projective image can be computed via class field theory. Secondly, weight 1
newforms over characteristic $p$ can be computed quickly, by the algorithms in
[14] or the $\mathbb{F}_{p}$ version of the Hecke stability method. These give
lower and upper bounds respectively on the dimension of a weight 1 newform
space of given level and character. A slower, deterministic computation of the
space is then only required if these bounds are not equal. This approach was
taken to compute all weight 1 forms up to level 1,500 as described in [8],
with the resulting data available at [1], and has since been performed up to
level 4,000 for the LMFDB (see [3]). From private discussion with Bill
Allombert, we are aware that such a computation up to level 5,000 has also
been performed.

This paper presents an improved version of the Hecke stability method to
compute all weight 1 newforms up to a given level. Improvements come from
utilising the twist-minimal trace formula presented in [10] to compute weight
2 forms, and the introduction of a method of lifting and verifying forms from
characteristic $p$, thus reducing the number of characteristic 0 calculations
which need to be performed. The outcome of these improvements is the
computation of all weight 1 newforms up to level 10,000, with this data
available at [9].

The structure of this paper is as follows. Section 2 covers the preliminary
theory used in our work, and presents our results. Section 3 constitutes a
proof of Theorem 1. Effectively, this is a proof that the proposed algorithm
does indeed produce a basis of any specified weight 1 newform space. Finally,
Section 4 discusses the practical computation, proving Theorem 2, that our
data up to level 10,000 is correct.

### Acknowledgements

This research is supported by EPSRC DTP EP/R513179/1 funding. I would like to
thank Min Lee, Andrew Booker and Jonathan Bober for their guidance and
suggestions with this work. I would also like to thank Andrew Sutherland for
providing the dihedral data.

##  2\. Computation of weight 1 forms

###  2.1. Preliminary theory

For any $N\in\mathbb{N}$, we define the congruence subgroup
$\Gamma_{0}(N)\subset\operatorname{SL_{2}}2(\mathbb{Z})$ by the following
condition:

$\Gamma_{0}(N)=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{SL_{2}}2(\mathbb{Z}):N\mid c\right\\}.$ (2)

Let $\chi$ be a level $N$ Dirichlet character. That is, a homomorphism:

$\chi:\left(\mathbb{Z}/N\mathbb{Z}\right)^{*}\rightarrow\mathbb{C},$ (3)

where the domain is extended to all $\mathbb{Z}$ by setting $\chi(n)=0$
whenever $(n,N)>1$. The conductor of $\chi$ is denoted $\mathfrak{f}(\chi)$.
We define $\chi(\gamma)$ for $\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma_{0}(N)$ by
$\chi(\gamma):=\chi(d)=\overline{\chi(a)}$.

The group $\operatorname{SL_{2}}2(\mathbb{R})$ acts on the Poincaré upper half
plane $\mathbb{H}=\\{z\in\mathbb{C}:\Im(z)>0\\}$ by Möbius transformations:

$\gamma
z=\frac{az+c}{cz+d},\;\;\;\;\;\;\;\forall\;\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{SL_{2}}2(\mathbb{R}).$ (4)

Fixing $k\in\mathbb{N}$, we define the weight $k$ slash action of
$\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{SL_{2}}2(\mathbb{R})$ on any complex-valued
function $f$ as:

$f|_{k}\gamma(z)=(cz+d)^{-k}f(\gamma z).$ (5)

If there exists a level $N$ Dirichlet character $\chi$ such that for all
$\gamma\in\Gamma_{0}(N)$ we have:

$f|_{k}\gamma(z)=\chi(\gamma)f(z),$ (6)

then we say that $f$ is weakly modular of weight $k$, level $N$ and character
$\chi$.

The set $\mathbb{Q}\cup\\{i\infty\\}$ is called the _cusps_ of
$\operatorname{SL_{2}}2(\mathbb{Z})$. It is the
$\operatorname{SL_{2}}2(\mathbb{Z})$-orbit of $i\infty$ under Möbius
transformations. If a function $f(z)$ is polynomially bounded as $z$ tends to
any cusp, then we say $f$ is ‘holomorphic at the cusps.’ This is equivalent to
requiring $f|_{k}\gamma(z)$ be polynomially bounded as
$\Im(z)\rightarrow\infty$ for all
$\gamma\in\operatorname{SL_{2}}2(\mathbb{Z})$.

Fix $k,N\in\mathbb{N}$ and let $\chi$ be a level $N$ Dirichlet character. A
weight $k$, level $N$, character $\chi$ modular form is a complex-valued
function $f$ which has the following properties:

  1. (1)

$f$ is holomorphic on $\mathbb{H}$.

  2. (2)

$f$ is holomorphic at the cusps.

  3. (3)

$f$ is weakly modular with weight $k$, level $N$ and character $\chi$.

One can immediately construct examples of modular forms, called Eisenstein
series, as follows. Fix a weight $k$, and let $\chi_{1}$ and $\chi_{2}$ be
Dirichlet characters level $N_{1}$ and $N_{2}$ respectively such that
$\chi_{1}\overline{\chi_{2}}(-1)=(-1)^{k}$. For $\Re(s)>1$ we define the
series:

$E_{\chi_{1}}^{\chi_{2}}(z;s)=\sum_{(c,d)=1}\frac{\Im(N_{2}z)^{s}\chi_{1}(c)\chi_{2}(d)}{|cN_{2}z+d|^{2s-k}(cN_{2}z+d)^{k}}$
(7)

If the weight $k$ is at least 3 then $E_{\chi_{1}}^{\chi_{2}}(z;k/2)$ is a
level $N_{1}N_{2}$ modular form with character $\chi_{1}\overline{\chi_{2}}$.
For weights 1 and 2 we obtain Eisenstein series by analytic continuation, and
in all cases the resulting modular form is written
$E_{\chi_{1}}^{\chi_{2}}(z)$. See [27] for details, including the derivation
of explicit expressions for Fourier coefficients. For any given Eisenstein
series $f$, there exists some cusp $c$ such that $f$ is not vanishing as
$z\rightarrow c$.

In contrast, if $f$ is a modular form such that $f(z)\rightarrow 0$ as $z$
approaches any cusp then we say that $f$ is a cusp form. All modular forms
(respectively cusp forms) of a given weight, level and character constitute a
vector space which we denote $\mathcal{M}_{k}(N,\chi)$ (respectively
$\mathcal{S}_{k}(N,\chi)$). The subspace of modular forms spanned by
Eisenstein series is denoted $\mathcal{E}_{k}(N,\chi)$, and any given modular
form space $\mathcal{M}_{k}(N,\chi)$ decomposes as
$\mathcal{S}_{k}(N,\chi)\oplus\mathcal{E}_{k}(N,\chi)$. Given two modular
forms $f$ and $g$, with at least one a cusp form, the Petersson inner product
is defined as:

$\langle
f,g\rangle_{N}=\int_{z=x+iy\in\Gamma_{0}(N)\backslash\mathbb{H}}f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^{2}}.$
(8)

With this inner product, the cusp form space of a given weight, level and
character is a Hilbert space and is orthogonal to the Eisenstein space. From
the definition of modularity, we see that if $f(z)\in\mathcal{S}_{k}(N,\chi)$
then $f(bz)\in\mathcal{S}_{k}(M,\chi)$ whenever $bN\mid M$. The subspace of
$\mathcal{S}_{k}(N,\chi)$ spanned by forms arising in this manner from levels
less than $N$ is called the oldform space and is denoted
$\mathcal{S}_{k}^{\text{old}}(N,\chi)$. The orthogonal complement of
$\mathcal{S}_{k}^{\text{old}}(N,\chi)$ with respect to the Petersson inner
product is called the newform space and is denoted
$\mathcal{S}_{k}^{\text{new}}(N,\chi)$. Consequently, any cusp form space
further decomposes into lifts from newform spaces. This decomposition was
presented and studied in [4].

Modular forms are completely described by their Fourier expansions:

$f(z)=\sum_{n\geq 0}a_{n}e^{2\pi inz}.$ (9)

Note that $f\in\mathcal{S}_{k}(N,\chi)$ implies that $a_{0}=0$. Let $f_{1}$
and $f_{2}$ be modular forms of the same weight, level and character. The
Sturm bound is a non-negative integer $B$ such that if $f_{1}\neq f_{2}$ then
the truncated Fourier expansions of $f_{1}$ and $f_{2}$ up to the $B$-th
coefficient must also differ. The Sturm bound is given by:

$B(\mathcal{M}_{k}(N,\chi))=\left\lfloor\frac{mk}{12}\right\rfloor,$ (10)

where $m$ is the index:

$m=\left[\operatorname{SL_{2}}2(\mathbb{Z}):\Gamma_{0}(N)\right]=N\prod_{p\mid
N}\left(1+\frac{1}{p}\right).$ (11)

This bound holds regardless of character (see [24, Corollary 9.20] for the
weight 2 case and [8, Lemma 5] for the weight 1 case). As such, computation of
Fourier coefficients of a modular form up to this bound identifies the form,
and a matrix of coefficients of basis elements, truncated at the Sturm bound,
will have full rank.

In [27] we find the following construction of basis elements of the Eisenstein
space, along with the Fourier expansions of these elements.

######  Definition 2.1.

Fix a level $N$ and Dirichlet character $\chi$. Let
$\mathcal{E}_{1}^{\text{basis}}(N,\chi)$ be the set of Eisenstein series
$E_{\chi_{1}}^{\chi_{2}}(z)$ of weight 1 satisfying the criteria:

  * •

$\chi_{2}(-1)=1$,

  * •

$\mathfrak{f}(\chi_{1})\mathfrak{f}(\chi_{2})\mid N$,

  * •

$\left(\mathfrak{f}(\chi_{1}),\frac{N}{\mathfrak{f}(\chi_{1})}\right)\mid\frac{N}{\mathfrak{f}(\chi)}$,

  * •

$\chi_{1}\overline{\chi_{2}}=\chi$.

The set of $E_{\chi_{1}}^{\chi_{2}}(bz)$ for any
$E_{\chi_{1}}^{\chi_{2}}(z)\in\mathcal{E}_{1}^{\text{basis}}(N,\chi)$ and
$b\mid\frac{N}{\mathfrak{f}(\chi_{1})\mathfrak{f}(\chi_{2})}$ is a basis for
$\mathcal{E}_{1}(N,\chi)$. Let $E_{\chi_{1}}^{\chi_{2}}(z)=\sum_{n\geq
0}a_{n}e^{2\pi inz}$ be any such Eisenstein series, then $a_{n}$ is given by:

$a_{n}=\begin{cases}\sum_{d\mid n}\chi_{1}(d)\chi_{2}(n/d)&\text{if }n>0,\\\
\frac{-1}{2\mathfrak{f}(\chi)}\sum_{r<\mathfrak{f}(\chi)}r\chi(r)&\text{if
}n=0\text{ and }\chi_{2}=\mathbbm{1},\\\ 0&\text{else.}\end{cases}$ (12)

For certain newforms, the Fourier coefficients are completely determined by
the eigenvalues of Hecke operators, which we now define as in [16]. Let $f$ be
a weight $k$ modular form, and fix $n\in\mathbb{N}$ and a Dirichlet character
$\chi$. The $n$-th Hecke operator $T_{n}^{\chi}(f)$ is given by:

$T_{n}^{\chi}(f)=\frac{1}{n}\sum_{ad=n}\chi(a)a^{k}\sum_{b\text{ mod
}d}f\left(\frac{az+b}{d}\right).$ (13)

All Hecke operators stabilise the spaces $\mathcal{M}_{k}(N,\chi),$
$\mathcal{E}_{k}(N,\chi)$, $\mathcal{S}_{k}(N,\chi)$,
$\mathcal{S}_{k}^{\text{old}}(N,\chi)$ and
$\mathcal{S}_{k}^{\text{new}}(N,\chi)$. We call a cusp form an _eigenform_ if
it is an eigenfunction of all Hecke operators. In [4] it is shown that any
newspace $\mathcal{S}_{k}^{\text{new}}(N,\chi)$ has a basis of eigenforms.
These can be normalised such that $a_{1}=1$, whereupon the $n$-th Hecke
eigenvalue is $a_{n}$. It is therefore these Hecke eigenvalues which we aim to
compute.

Hecke relations (see [11, Proposition 10.2.5]) allow for the recovery of any
$a_{n}$ given knowledge of $a_{p}$ for relevant primes $p$. Define $a_{r}=0$
for any $r\not\in\mathbb{N}$. A normalised Hecke eigenform has $a_{1}=1$, and
for any $n>1$ with prime $p\mid n$, the $n$-th Fourier coefficient of the
eigenform is given by:

$a_{n}=a_{\frac{n}{p}}a_{p}-\chi(p)p^{2k-1}a_{\frac{n}{p^{2}}}$ (14)

Computation of the $p$-th Hecke eigenvalue of an arbitrary weight 1 eigenform
is, however, not an easy task. While a trace formula for all automorphic forms
of weight 1 can be generated, it is not possible to extract the holomorphic
contribution (see [15, Chapter 10, Section 4]). As such, computation of the
$a_{p}$ has been performed a number of indirect ways (see [11, Section 13.6.1]
for a review of three methods), with the most efficient being the _Hecke
stability method_ presented in [23]. This is performed as follows:

  1. (1)

Let $N$ and $\chi$ be the given level and character of the desired weight 1
newform space. Compute the space $S_{2}(N,\mathbbm{1})$.

  2. (2)

Divide this by any Eisenstein series $E\in\mathcal{E}_{1}(N,\overline{\chi})$.
Denote the result $\mathcal{M}_{1}^{*}(N,\chi)$.

  3. (3)

Find the maximal subspace of $\mathcal{M}_{1}^{*}(N,\chi)$ which is stable
under all Hecke operators. Denote this $\tilde{V}$.

  4. (4)

We have
$\mathcal{S}_{1}(N,\chi)\subset\tilde{V}\subset\mathcal{M}_{1}(N,\chi)$, and
so removing the contribution of Eisenstein series and oldforms from
$\tilde{V}$ gives $\mathcal{S}_{1}^{\text{new}}(N,\chi)$.

Practically, ‘computing a space’ means computing a matrix of basis
coefficients up to the Sturm bound. It is also shown in [23] that this method
works (under additional constraints) for computing weight 1 forms over
characteristic $p$. Computation over such fields is considerably faster than
computation over $\mathbb{Q}$, and given that newforms over $\mathbb{Q}$
project onto $\mathbb{F}_{p}$ as newforms, one might hope that only the
characteristic $p$ computation is needed. Unfortunately, there exist so-called
‘ethereal forms’ (see [14, Appendix A]) over characteristic $p$ which are not
the projection of forms over $\mathbb{Q}$. Thus, restricting solely to
computation over characteristic $p$ is an unreliable method for recovering
forms over $\mathbb{Q}$.

We now move onto Artin representations. Weight 1 eigenforms are associated in
an explicit fashion with Artin representations, and we use this association to
deduce important results about their Hecke eigenvalues. Let $G_{\mathbb{Q}}$
be the absolute Galois group
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

######  Definition 2.2.

An $n$-dimensional Artin representation $\rho$ is a continuous homomorphism.

$\rho:G_{\mathbb{Q}}\rightarrow\text{GL}_{n}(\mathbb{C}).$ (15)

We can compose $\rho$ with
$\det:\text{GL}_{n}(\mathbb{C})\rightarrow\mathbb{C}^{\times}$ to give the
‘determinant character’ $\chi$, a 1-dimensional Artin representation. Letting
$\iota\in G_{\mathbb{Q}}$ be complex conjugation, we say that $\rho$ is odd if
$\det(\rho(\iota))=-1$ and $\rho$ is even if $\det(\rho(\iota))=1$. An Artin
representation has the same parity as its determinant character.

Continuity implies that $\rho$ filters through a finite extension
$K/\mathbb{Q}$, and thus the image of $\rho$ under projection onto
$\text{PGL}_{n}(\mathbb{C})$ is a finite subgroup of
$\text{PGL}_{n}(\mathbb{C})$. We categorise 2-dimensional Artin
representations by their image under this projection, which must be either
cyclic, dihedral, or one of $A_{4},S_{4}$ or $A_{5}$. If $\rho$ is
irreducible, then the cyclic case is impossible. If the image of an
irreducible representation is not dihedral, then we say it is exotic.

We follow [21] to define the $L$-function attached to an Artin representation
$\rho$. Let $K$ be the minimal number field through which $\rho$ filters. For
any prime $p$, fix a prime $\mathfrak{p}$ lying above $p$, with inertia group
$I_{\mathfrak{p}}$. Let $V^{I_{\mathfrak{p}}}$ be the fixed subspace:

$V^{I_{\mathfrak{p}}}=\left\\{x\in\mathbb{C}^{n}:\sigma(x)=\sigma\;\;\forall\;\sigma\in\rho(I_{\mathfrak{p}})\right\\}.$
(16)

Let $Frob_{\mathfrak{p}}$ be any Frobenius element for the ideal
$\mathfrak{p}$. Then the determinant:

$\det\left(1-p^{-s}\rho|_{V^{I_{\mathfrak{p}}}}(Frob_{\mathfrak{p}})\right),$
(17)

is seen to be well-defined and irrespective of the choice of $\mathfrak{p}$.
We define the Artin $L$-function attached to $\rho$ by the Euler product:

$L(\rho,s)=\prod_{p}\det\left(1-p^{-s}\rho|_{V^{I_{\mathfrak{p}}}}(Frob_{\mathfrak{p}})\right)^{-1}.$
(18)

We state the association with eigenforms via this $L$-function. One can
construct an $L$-function from any eigenform $f$ by the product:

$L(f,s)=\sum_{n}\frac{a_{n}}{n^{s}}=\prod_{p\mid
N}\left(1-a_{p}p^{-s}\right)^{-1}\prod_{p\nmid
N}\left(1-a_{p}p^{-s}+\chi(p)p^{k-1-2s}\right)^{-1}.$ (19)

By [13, Theorem 4.1], all such $L$-functions for weight 1 eigenforms are
equivalent to $L$-functions arising from Artin representations, and thus all
weight 1 eigenforms have an associated Artin representation. In [17] it is
proven that the converse is also true. Consequently, we use the $L(\rho,s)$
construction to deduce information about $L(f,s)$ and in turn $a_{p}$.

We define the Satake parameters at $p$ as the eigenvalues of
$\rho|_{V^{I_{\mathfrak{p}}}}(Frob_{\mathfrak{p}})$. As these are all matrices
of finite order, Satake parameters are roots of unity. For an unramified
prime, the inertia group is trivial and so we have two (not necessarily
distinct) Satake parameters. For a ramified prime, we have zero or one satake
parameters, corresponding to the dimension of $V^{I_{\mathfrak{p}}}$.

The specific $L$-function association given in [13] includes that a prime $p$
is unramified in $K$ if and only if $p\nmid N$ where $N$ is the level of the
associated eigenform $f$. Thus we can equivalently define Satake parameters in
terms of the Fourier expansion of $f$:

  * •

For $p\nmid N$, the roots of $x^{2}-a_{p}x+\chi(p)$ are the Satake parameters
at $p$.

  * •

For $p\mid N$ with $a_{p}=0$, there are no Satake parameters at $p$.

  * •

For $p\mid N$ with $a_{p}\neq 0$, $a_{p}$ is the Satake parameter at $p$.

From [19, Theorem 3], we see that:

$p^{2}\mid N,\;\nu_{p}(\mathfrak{f}(\chi))<\nu_{p}(N)\iff a_{p}=0.$ (20)

Therefore, we can compute $a_{p}$ for any prime $p$ given $N$, $\chi$ and a
Satake parameter at $p$ (when one exists). As an immediate application, [19,
Theorem 3] shows that when $\nu_{p}(N)=1$ and $\chi_{p}=\mathbbm{1}$ we must
have $|a_{p}|=\sqrt{p}$, but as the Satake parameters are roots of unity, this
cannot be the case, and so $\mathcal{S}_{1}^{\text{new}}(N,\chi)=0$. In
particular, there are no non-zero weight 1 newforms for any level $N\equiv
2\pmod{4}$.

The subspaces of $\mathcal{S}_{1}^{\text{new}}(N,\chi)$ spanned by eigenforms
associated with Artin representations with dihedral or exotic projective image
are denoted $\mathcal{S}_{1}^{\text{dihedral}}(N,\chi)$ and
$\mathcal{S}_{1}^{\text{exotic}}(N,\chi)$ respectively. Computation of the
dimensions of $\mathcal{S}_{1}^{\text{dihedral}}(N,\chi)$, along with
projective images of forms in these spaces, can be performed efficiently via
class field theory, as explained in [7]. Such a computation, giving the
dimensions and specific projective images of all weight 1 dihedral forms, has
been performed up to level 40,000 and detailed in [5]. We assume this
information for our computation and so reduce our scope to the computation of
$\mathcal{S}_{1}^{\text{exotic}}(N,\chi)$.

###  2.2. Results

Our computation is primarily the result of improvements to the Hecke stability
method. Firstly, we describe an efficient computation of
$\mathcal{S}_{2}(N,\mathbbm{1})$ coming from the twist-minimal trace formula
given in [10].

######  Definition 2.3.

Fix a level $N\in\mathbb{N}$. A _twist pair_ $\langle M,\psi\rangle$ is a
tuple of $M\mid N$ and Dirichlet character $\psi$ satisfying the following
conditions:

  * •

For all $p$ with $\psi_{p}\neq\mathbbm{1}$,
$\nu_{p}(N)=2\nu_{p}(\mathfrak{f}(\psi))$;

  * •

For all $p>2$ either $\nu_{p}(M)=\nu_{p}(N)-\nu_{p}(\mathfrak{f}(\psi))$, or
$\nu_{p}(M)=0$ and $\operatorname{Ord}(\psi_{p})=2$;

  * •

If $2\mid\nu_{2}(N)$ and $\nu_{2}(N)>3$ then
$\nu_{2}(M)\in\left\\{\frac{\nu_{2}(N)}{2}-1,\nu_{2}(N)-1,\nu_{2}(N)-2\right\\}$;

  * •

If $\psi_{2}=\mathbbm{1}$ then $\nu_{2}(M)=\nu_{2}(N)$.

Note that if $\psi_{1}$ and $\psi_{2}$ are Galois-conjugate characters, and
$\langle M,\psi_{1}\rangle$ is a twist pair, then so is $\langle
M,\psi_{2}\rangle$. We can thus associate any Galois orbit with a Dirichlet
character $\Psi$ defined over the cyclotomic field $\mathbb{Q}(\zeta_{r})$
where $r$ is the order of the orbit. Any twist pair $\langle M,\psi\rangle$
with $\psi$ in the orbit then arises as an embedding of $\langle
M,\Psi\rangle$ into $\mathbb{C}$.

For any twist-pair $\langle M,\psi\rangle$ we define the trace form
$\mathcal{T}_{M,\psi}$ by the Fourier expansion $\sum_{n}a_{n}e^{2\pi izn}$
where the $a_{n}$ are given as follows. If $(n^{2},N)$ is not squarefree then
$a_{n}=0$. Otherwise:

$a_{n}=(C_{1}-C_{2}-C_{3}+C_{4})\prod_{\begin{subarray}{c}p|\mathfrak{f}(\psi)\\\
p\nmid M\end{subarray}}\psi_{p}(n),$ (21)

with:

$\begin{split}C_{1}&=\frac{\underset{n\text{ is
square}}{\delta}}{12}\prod_{p|M}\begin{cases}p^{e}+p^{e-1}&\text{if }s=e,\\\
\frac{\phi(\lceil p^{e-2}\rceil)(p-1)}{1+\underset{\begin{subarray}{c}2|e\\\
p>2\end{subarray}}{\delta}}(1+\underset{e>1}{\delta}p-\underset{e=2}{\delta}2)&\text{else.}\end{cases}\\\
C_{2}&=\sum_{t^{2}<4n}\frac{h(d)}{w(d)}\prod_{\begin{subarray}{c}p|\ell\\\
p\nmid
M\end{subarray}}S_{p}(1,\mathbbm{1},t,n)\prod_{p|M}S_{p}^{\min}(p^{e},\psi_{p}^{2},t,n)_{\overline{\psi_{p}}}\\\
C_{3}&=\sum_{\begin{subarray}{c}d|n\\\
d\leq\sqrt{n}\end{subarray}}^{\prime}d\prod_{p|M}\begin{cases}\Re\left(\psi\left(\frac{d^{2}}{n}\right)\right)&\text{if
}s=e,\\\ 0&\text{else.}\end{cases}\\\
C_{4}&=\underset{\psi^{2}=\mathbbm{1}}{\delta}\mu(M)\prod_{\begin{subarray}{c}p|n\\\
p\nmid M\end{subarray}}\sigma(p^{\nu_{p}(n)})\\\ \end{split}$ (22)

In all terms, $s=\nu_{p}(\mathfrak{f}(\psi^{2}))$, $e=\nu_{p}(M)$ and $\delta$
is the characteristic function of the condition underneath it. In $C_{1}$,
$\phi(a)$ is Euler’s totient function, the size of
$(\mathbb{Z}/a\mathbb{Z})^{*}$. In $C_{2}$, we define $d$ as the fundamental
discriminant of $t^{2}-4n$, with $\ell\in\mathbb{N}$ such that
$t^{2}-4n=d\ell^{2}$. The functions $h(d)$ and $w(d)$ are the class number and
roots of unity of a quadratic extension with fundamental discriminant $d$. In
$C_{3}$, the dash on the sum indicates that if $d=\sqrt{n}$ an extra factor of
$\frac{1}{2}$ is present, and we define $\Re(\chi(x))$ as
$\chi(x)+\chi\left(x^{-1}\right)$ when $x$ is invertible, and $0$ otherwise.
The functions $\mathcal{S}_{p}$ and $\mathcal{S}_{p}^{\min}$ present in
$C_{2}$ are defined as follows. Set $v=\nu_{p}(t^{2}-4n)$, and let
$\left(\frac{\cdot}{p}\right)$ denote the Kronecker symbol. For $p>2$ we
define:

$S_{p}^{\min}(p^{e},\psi_{p},t,n)_{\overline{\psi}}=\begin{cases}2p^{\nu_{p}(\ell)}+\left(1-\left(\frac{d}{p}\right)\right)\frac{2p^{\nu_{p}(\ell)}-p^{e}-p^{e-1}}{p-1}&\text{if
}s>0,v\geq 2e-1\\\ \\\
\underset{\left(\frac{d}{p}\right)=1}{\delta}p^{\nu_{p}(\ell)}\Re\left(\psi\left(\frac{t(t+u)}{2n}-1\right)\right)&\text{if
}s>0,v<2e-1\\\ \\\
\left(\left(\frac{d}{p}\right)-1\right)\left(\frac{n}{p}\right)&\text{if
}s=0,p|\mathfrak{f}(\psi)\\\ \\\ \underset{\begin{subarray}{c}e=1\\\ \text{or
}\left(\frac{n}{p}\right)=1\end{subarray}}{\delta}\left(1-\left(\frac{d}{p}\right)\right)\frac{p^{e-3}}{(2,e)}\\\
\cdot(\underset{e>2}{\delta}+p(\underset{e=2}{\delta}+\underset{\begin{subarray}{c}2|e\\\
v=e-2\end{subarray}}{\delta}-\underset{v\geq e-1}{\delta}p))&\text{if }v\geq
e-2,p\nmid\mathfrak{f}(\psi)\\\ \\\ 0&\text{else}\end{cases}$ (23)

For $p=2$ with $s=e$ we define:

$S_{2}^{\min}(2^{e},\psi^{2},t,n)_{\overline{\psi}}=\begin{cases}(1-\underset{v=2e}{\delta}2)\\\
\cdot\left((2^{\lfloor\frac{v}{2}\rfloor+1}-3\cdot
2^{e-1})(1-\left(\frac{d}{2}\right))+\underset{2\nmid
d}{\delta}2^{\nu_{2}(\ell)+1}\right)&\text{if }v\geq 2e\\\
\underset{\left(\frac{d}{2}\right)=1}{\delta}2^{\nu_{2}(\ell)}\Re\left(\psi\left(\frac{t(t+u)}{2n}-1\right)\right)&\text{if
}v<2e-1\\\ \end{cases}$ (24)

For $p=2$ with $2|\mathfrak{f}(\psi)$ but $s<e$ we define:

$S_{2}^{\min}(2^{e},\mathbbm{1},t,n)_{\overline{\psi}}=\left(1-\left(\frac{d}{2}\right)\right)\lceil
2^{e-3}\rceil\begin{cases}-3&\text{if }v>e\geq 3\\\
\psi(n)(2(-1)^{d}-1)&\text{if }v\in\\{e,e-1\\},e\geq 3,s=0\\\
(-1)^{e}+2&\text{if }s>0,v=e\\\ 1-2(-1)^{d}&\text{if }s>0,v=e-1\\\
\psi(n)\left(\underset{\begin{subarray}{c}e=2\\\
v=0\end{subarray}}{\delta}\frac{3}{2}-1\right)&\text{if }e\in\\{1,2\\}\\\
0&\text{else}\end{cases}$ (25)

In all these cases, $u\equiv\ell\sqrt{d}$ is defined mod $p^{e+2}$ if $p=2$
and mod $p^{e}$ otherwise. Finally, we define:

$S_{p}(1,\mathbbm{1},t,n)=p^{\nu_{p}(\ell)}+\left(1-\left(\frac{d}{p}\right)\right)\frac{p^{\nu_{p}(\ell)}-1}{p-1}.$
(26)

We can construct a polynomial traceform $\mathcal{T}_{M,\Psi}$ by this
formula, evaluating the character $\Psi$ associated with the Galois orbit of
some twist pair $\langle M,\psi\rangle$.

######  Lemma 1.

For any polynomial traceform $\mathcal{T}_{M,\Psi}$ associated to the Galois
orbit containing $\langle M,\psi\rangle$, denote $r=\operatorname{Ord}(\psi)$
and let $\beta$ be the Sturm bound at weight 2 and level $M$. The first
coefficient of $\mathcal{T}_{M,\Psi}$ is a non-negative integer, which we
denote $d$. There exist $A$ and $B$ finite subsets of $\mathbb{N}$, with
$a\leq\frac{r}{2}$ for all $a\in A$, $b\leq\beta$ for all $b\in B$, and
$|B|=d$, such that embeddings given by $\zeta_{r}\rightarrow e^{2\pi ia/r}$ of
elements $T_{b}\mathcal{T}_{M,\Psi}$ across all $a\in A,b\in B$ and Galois
orbits of twist pairs $\langle M,\psi\rangle$ constitutes a basis for
$\mathcal{S}_{2}^{\text{new}}(N,\mathbbm{1})$.

The sets $A$ and $B$ are difficult to write down directly, but in practice,
easy to generate iteratively for any given case. To find $B$ we check each
possible natural number $b$ in turn, appending to the set if
$T_{b}\mathcal{T}_{M,\Psi}$ is not some linear combination of the established
$T_{b^{\prime}}\mathcal{T}_{M,\Psi}$ for $b^{\prime}\in B$, until we have $d$
linearly independent elements.

To find $A$ we proceed as follows. We say that $\psi_{1}$ and $\psi_{2}$ are
twist-equivalent if for every prime $p$ we have
$\psi_{1,p}\in\\{\psi_{2,p},\overline{\psi_{2,p}}\\}$. For a twist pair
$\langle M,\Psi\rangle$ we check each possible natural number $a$ in turn,
appending to $A$ if there does not already exist a twist pair $\langle
M,\Psi^{\prime}\rangle$ with $Ord(\Psi)=Ord(\Psi^{\prime})=r$ and embedding
$a^{\prime}$ such that the character $\psi$ given by embedding $\Psi$ into
$\mathbb{C}$ by $\zeta_{r}\rightarrow e^{2\pi ia/r}$ is twist-equivalent to
the character $\psi^{\prime}$ given by embedding $\Psi^{\prime}$ into
$\mathbb{C}$ by $\zeta_{r}\rightarrow e^{2\pi ia^{\prime}/r}$.

Given the efficient computation of arbitrary coefficients of a basis for
$\mathcal{S}_{2}^{\text{new}}(N,\mathbbm{1})$, we present an algorithm for
computing $\mathcal{S}_{1}^{\text{new}}(N,\chi)$ from this basis, assuming
knowledge of the dimensions and projective images of
$\mathcal{S}_{1}^{\text{dihedral}}(N,\chi)$, and with the set
$\mathcal{E}_{1}^{\text{basis}}(N,\chi)$ as in Definition 2.1. This is
essentially the Hecke stability method with practical improvements.

  1. (1)

Let $N$ and $\chi$ be the given level and character of the desired weight 1
newform space. Let $D$ be the LCM of all orders of projective images of Artin
representations associated with eigenbasis elements for
$\mathcal{S}_{1}^{\text{dihedral}}(N^{\prime},\chi)$ where $N^{\prime}$ ranges
across all levels satisfying $\mathfrak{f}(\chi)\mid N^{\prime}\mid N$, and
let $H=2\operatorname{LCM}(60,\phi(N),D)$, where $\phi$ is Euler’s totient
function. Fix the least primes $p$ and $q$ satisfying $p\nmid N$ and $q\equiv
1\pmod{H}$.

  2. (2)

Let $d=\dim(\mathcal{S}_{2}(N,\mathbbm{1}))$. Compute a $m\times d$ basis
matrix $M$ for $\mathcal{S}_{2}(N,\mathbbm{1})$ using Lemma 1, where $m$ is at
least the Sturm bound of $\mathcal{S}_{1}(N,\chi)$ and is such that the
submatrix of the first $m/p$ rows has rank $d$.

  3. (3)

Fix a ring homomorphism
$\sigma:\mathbb{Z}[\zeta_{H}]\rightarrow\mathbb{F}_{q}$ whose restriction to
$\langle\zeta_{H}\rangle$ is injective, and use this to project $M$ and
$\mathcal{E}_{1}^{\text{basis}}(N,\overline{\chi})$ onto $\mathbb{F}_{q}$.

  4. (4)

Let $M/E$ for an Eisenstein series $E$ be the $m\times d$ matrix given by the
leading $m$ coefficients of each basis element in $M$ divided by $E$. Working
over $\mathbb{F}_{q}$, find $M^{*}$, the intersection of $M/E$ across all
$E\in\mathcal{E}_{1}^{\text{basis}}(N,\overline{\chi})$.

  5. (5)

Find $\tilde{V}$, the maximum $T_{p}$-stable subspace of $M^{*}$. Diagonalise
$\tilde{V}$, removing the contribution from
$\mathcal{S}_{1}^{\text{old}}(N,\chi)$ in the process. The details of this
diagonalisation will be given in the next section.

  6. (6)

For each remaining element $\tilde{f}$ in $\tilde{V}$ and each prime $p\leq
m$, compute a (non-zero) root $\alpha_{p,\mathbb{F}_{q}}$ of
$x^{2}-a_{p}x+\chi(p)$ over $\mathbb{F}_{q}$. If there is some $p$ where this
polynomial doesn’t split over $\mathbb{F}_{q}$ then discard $\tilde{f}$.

  7. (7)

For all primes $p\leq m$ let $\alpha_{p,\mathbb{Q}}$ be the root of unity
which is the preimage of $\alpha_{p,\mathbb{F}_{q}}$ under $\sigma$. Define a
lifted $f$ over $\mathbb{Z}[\zeta_{H}]$ by setting
$a_{p}=\alpha_{p,\mathbb{Q}}+\chi(p)\overline{\alpha_{p,\mathbb{Q}}}$ and
applying Hecke relations to recover all $a_{n}$.

  8. (8)

Multiply $f$ over $\mathbb{Q}$ by any element in
$\mathcal{E}_{1}^{\text{basis}}(N,\overline{\chi})$. If the result is in
$\mathcal{S}_{2}(N,\mathbbm{1})$ then include $f$ in the output.

######  Theorem 1.

The output of the preceding algorithm is an eigenbasis for
$\mathcal{S}_{1}^{\text{new}}(N,\chi)$.

The computation of these eigenbases can be performed for each level in turn,
so that the computed basis of $\mathcal{S}_{2}(N,\mathbbm{1})$ can be reused
for each character, and the elements in $\mathcal{S}_{1}^{\text{new}}(M,\chi)$
can be lifted to $\mathcal{S}_{1}^{\text{old}}(N,\chi)$ for every $M\mid N$ in
step 5. Not every character and level needs to be computed, which motivates
the following definition.

######  Definition 2.4.

Fix $N\in\mathbb{N}$ and a level $N$ Dirichlet character $\chi$. We say this
pair is _admissible_ if the following criteria are all met:

  * •

The character $\chi$ is odd;

  * •

For any $p$ with $\nu_{p}(N)=1$, $\chi_{p}\neq\mathbbm{1}$;

  * •

For any $p\mid N$ with $\nu_{p}(N)=\nu_{p}(\mathfrak{f}(\chi))$,
$\operatorname{Ord}(\chi_{p})\in\left\\{2,3,4,5\right\\}$;

  * •

For any two such primes $p$ and $q$, $\operatorname{Ord}(\chi_{p}\chi_{q})\neq
20$.

######  Lemma 2.

Fix $N_{\max}\in\mathbb{N}$, the maximum level to compute all weight 1 exotic
forms up to. For any given $N\leq N_{\max}$ and level $N$ Dirichlet character
$\chi$, let $M$ be the least multiple of $N$ such that $M$ and $\chi$ are
admissible. If $M>N_{\max}$, or if no such $M$ exists, then
$\mathcal{S}_{1}^{\text{new}}(N,\chi)$ does not need to be computed. Computing
all other $\mathcal{S}_{1}^{\text{new}}(N,\chi)$ with $N\leq N_{\max}$
provides the dimensions (and Fourier coefficients) of all exotic forms with
level at most $N_{\max}$.

We performed such a computation up to level $10,000$. This took 832 minutes on
192 processors, implemeted in PARI (see [26]) and performed in parallel using
Parallel (see [25]).

######  Theorem 2.

The dimensions of all spaces of newforms
$\mathcal{S}_{1}^{\text{new}}(N,\chi)$ for $N$ up to level 10,000, and the
coefficients of all newforms in spaces where
$\mathcal{S}_{1}^{\text{exotic}}(N,\chi)\neq 0$, up to at least the Sturm
bound, are as given at [9].

##  3\. Proof of Theorem 1

The formula for $\mathcal{T}_{M,\chi}$ preceding Lemma 1 is a specific
application of the twist-minimal trace formula from [10]. Let
$f\in\mathcal{S}_{k}^{\text{new}}(N,\chi)$ be a newform with Fourier expansion
$\sum a_{n}e^{2\pi inz}$. The twist of $f$ by $\psi$ is defined as:

$f_{\psi}(z)=\sum_{n>0}a_{n}\psi(n)e^{2\pi inz}.$ (27)

This is a cusp form of potentially different level to $N$, and character
$\chi\psi^{2}$. As in the decomposition of cusp forms into lifts from spaces
of newforms, we define spaces of twist-minimal forms as the orthogonal
complement to the space spanned by forms arising as twists from lower levels.
Spaces of newforms consequently decompose into twists from twist-minimal
spaces, denoted $\mathcal{S}_{k}^{\min}(N,\chi)$. This decomposition only
includes those twist-minimal spaces with twist-minimal characters, defined as
follows:

######  Definition 3.1.

Let $\chi$ be a level $N$ Dirichlet character. We say that $\chi$ is twist-
minimal if for every $p\mid N$ at least one of the following holds:

  * •

$\chi_{p}$ is primitive;

  * •

$p>2$ and $\chi_{p}=\mathbbm{1}$;

  * •

$p>2$ and $\text{Ord}(\chi_{p})=2^{\nu_{2}(p-1)}$;

  * •

$p=2$ and $\nu_{p}(\mathfrak{f}(\chi))=\lfloor\frac{\nu_{p}(N)}{2}\rfloor$;

  * •

$p=2$, $\nu_{p}(\mathfrak{f}(\chi))=2$, $\nu_{p}(N)>3$ and $2\nmid\nu_{p}(N)$;

  * •

$p=2$, $\chi_{p}=\mathbbm{1}$, and $2\nmid\nu_{p}(N)$ or $\nu_{p}(N)=2$.

The definition of twist pairs in Definition 2.3 are seen to be such that
$\langle M,\psi\rangle$ is a twist pair if and only if $\overline{\psi^{2}}$
is a level $M$ twist-minimal character. An explicit trace formula for twist-
minimal spaces with twist-minimal character is given in [10, Theorem 1].

If
$\mathcal{S}_{2}^{\min}(M_{1},\overline{\psi_{1}^{2}})_{\psi_{1}}=\mathcal{S}_{2}^{\min}(M_{2},\overline{\psi_{2}^{2}})_{\psi_{2}}$
then we say the pairs $\langle M_{1},\psi_{1}\rangle$ and $\langle
M_{2},\psi_{2}\rangle$ are twist-equivalent. We see from [10, Theorem 2] that
a basis for $\mathcal{S}_{2}^{\text{new}}(N,\mathbbm{1})$ is given by bases of
$\mathcal{S}_{2}^{\min}(M,\overline{\psi^{2}})_{\psi}$ where $M$ and $\psi$
range across twist pairs, excepting those pairs which are twist-equivalent.

Following [10, Theorem 1] we see that the $n$-th coefficient of the trace form
of $\mathcal{S}_{2}^{\min}(M,\overline{\psi^{2}})_{\psi}$ is given by:

$a_{n}(C_{1}-C_{2}-C_{3}+C_{4})\prod_{p\mid\mathfrak{f}(\psi)}\psi_{p}(n),$
(28)

where $C_{i}$ are the components of the twist minimal trace formula evaluated
for $\mathcal{S}_{2}^{\min}(M,\overline{\psi^{2}})$. Taking the evaluation of
$\psi$ for factors $p\mid M$ into the sum gives the following expression for
$C_{1}$:

$C_{1}=\frac{\psi(n)\overline{\psi^{2}(\sqrt{n})}}{12}\prod_{p\mid
M}\begin{cases}p^{e}+p^{e-1}&\text{if }s=e,\\\ \frac{\phi(\lceil
p^{e-2}\rceil)(p-1)}{1+\underset{\begin{subarray}{c}2\mid e\\\
p>2\end{subarray}}{\delta}}+\underset{e>1}{\delta}p+\underset{e=2}{\delta}(2s-2))&\text{else,}\end{cases}$
(29)

where $s=\nu_{p}(\mathfrak{f}(\psi^{2}))$, $e=\nu_{p}(M)$ and
$\psi^{2}(\sqrt{n})=0$ if $n$ is not square. We see from the definition of
twist pairs that $e=2$ and $s\neq e$ implies $s=0$, and the evaluations of
$\psi$ cancel, giving $C_{1}$ as in (22). The remaining terms come from
similar direct evaluations of the twist-minimal trace formula.

We note that each term in the formula is only dependent on evaluations of
$\psi_{p}$, its level and conductor, and whether or not $\psi_{p}$ is
quadratic. These are equal for every character in the Galois orbit of $\psi$,
and therefore any $\mathcal{T}_{M,\psi}$ is the embedding of the polynomial
traceform $\mathcal{T}_{M,\Psi}$ attached to the Galois orbit containing
$\psi$.

As Galois conjugate spaces are the same dimension, the first coefficient of
$\mathcal{T}_{M,\Psi}$ is an integer $d$ giving this dimension. As each
$T_{m}\mathcal{T}_{M,\psi}$ is also conjugate, we can produce (by [10, Theorem
2]) a rank $d$ matrix consisting of leading coefficients of
$T_{m}\mathcal{T}_{M,\Psi}$ for various $m$, whereupon a basis for
$\mathcal{S}_{2}^{\min}(M,\overline{\psi^{2}})_{\psi}$ is given by embedding
this matrix into $\mathbb{C}$ with the map which sends $\Psi$ to $\psi$.

Taking this computation across all twist-pairs will count twists from twist-
equivalent spaces multiple times, and so we use the following lemma to remove
duplicates.

######  Lemma 3.

Suppose $\chi$ is twist-equivalent to $\psi$. Let $\chi^{\prime}$ be some
character in the Galois orbit of $\chi$. Then there exists some
$\psi^{\prime}$ in the Galois orbit of $\psi$ which is twist-equivalent to
$\chi^{\prime}$.

###### Proof.

For each prime we have $\chi_{p}^{\prime}=\chi_{p}^{n}$. Define
$\psi_{p}^{\prime}=\psi_{p}^{n}$. As $\chi$ is twist-equivalent to $\psi$ we
must have $\psi_{p}\in\\{\chi_{p},\overline{\chi_{p}}\\}$. Thus,
$\psi_{p}^{\prime}\in\\{\chi_{p}^{n},\overline{\chi_{p}^{n}}\\}$ for all $p$,
and so $\psi^{\prime}$ is twist-equivalent to $\chi^{\prime}$. ∎

This means that twist-equivalence is an equivalence relation on Galois orbits
as well as individual characters. If we compute $\mathcal{T}_{M,\Psi}$ for all
Galois orbits, extract a full-rank basis matrix from this, and embed the
results into all $e^{2\pi ia/r}$ $a\in A$, the set of maximal distinct
embeddings, then we recover a basis for
$\mathcal{S}_{2}^{\text{new}}(N,\mathbbm{1})$ without duplication.

The remainder of this section constitutes the proof of Theorem 1.

######  Lemma 4.

Let $M^{*}$ be the intersection of $\mathcal{S}_{2}(N,\mathbbm{1})/E$ for each
$E\in\mathcal{E}_{1}^{\text{basis}}(N,\overline{\chi})$. Fix a prime $p\nmid
N$. Then the maximal $T_{p}$-stable subspace of $M^{*}$, denoted $\tilde{V}$,
is $\mathcal{S}_{1}(N,\chi)$.

###### Proof.

In [23] it is proven that, for any Eisenstein series
$E\in\mathcal{E}_{1}(N,\overline{\chi})$ and $p\nmid N$, the maximal
$T_{p}$-stable subspace of $\mathcal{S}_{2}(N,\mathbbm{1})/E$ is a subspace of
$\mathcal{M}_{1}(N,\chi)$ containing $\mathcal{S}_{1}(N,\chi)$. Thus:

$\mathcal{S}_{1}(N,\chi)\subset\tilde{V}\subset\mathcal{M}_{1}(N,\chi).$ (30)

In [27], $\mathcal{E}_{1}^{\text{basis}}(N,\chi)$, along with lifts thereof,
is shown to span $\mathcal{E}_{1}(N,\chi)$. For any
$E\in\mathcal{E}_{1}^{\text{basis}}(N,\overline{\chi})$, the existence of some
cusp at which $E$ is non-zero gives:

$E(Bz)\overline{E(z)}\not\in\mathcal{S}_{2}(N,\mathbbm{1}).$ (31)

Consequently, $M^{*}\cap\mathcal{E}_{1}(N,\chi)=0$ and so
$\tilde{V}=\mathcal{S}_{1}(N,\chi)$. ∎

We now show that any form over $\mathbb{Q}$ will project onto $\mathbb{F}_{q}$
for a carefully chosen prime $q$ in such a way that the original form can be
recovered. We first note that $\tilde{V}$ can be computed over
$\mathbb{F}_{q}$, and so we acquire a space containing the projection of
$\mathcal{S}_{1}(N,\chi)$.

######  Lemma 5.

Define $H$ as in Theorem 1. Fix a prime $q$ such that $q\equiv 1\pmod{H}$ and
$q\nmid N$. Fix a projection $\mathbb{Z}[\zeta_{H}]\rightarrow\mathbb{F}_{q}$,
then the computation of $\mathcal{S}_{2}(N,\mathbbm{1})$,
$\mathcal{E}_{1}^{\text{basis}}(N,\overline{\chi})$, and $\tilde{V}$ are all
well-defined over $\mathbb{F}_{q}$.

###### Proof.

First, we show that all coefficients in the basis of
$\mathcal{S}_{2}(N,\mathbbm{1})$ are in $\mathbb{Z}[\zeta_{H}]$. This comes
from the fact that trace form coefficients for $\mathcal{S}_{k}(N,\chi)$ are
in $\mathbb{Z}[\chi]$ (see [18, Theorem 28.4]). By [10, (30) and (34)], we can
write $\text{Tr}T_{n}|\mathcal{S}_{k}^{\min}(N,\chi)$ as a combination of
elements $\text{Tr}T_{n}|\mathcal{S}_{k}^{\min}(N^{\prime},\psi)$, where all
$\psi$ have order dividing $\phi(N)$.

The Fourier expansion of Eisenstein series in (12) is similarly given in terms
of evaluations of characters with conductor dividing $N$ and thus order
dividing $\phi(N)$, and so are in $\mathbb{Z}[\zeta_{H}]$. Finally, the vector
representing action by the $T_{p}$ operator on the Fourier expansion of any
modular form has coefficients in $\mathbb{Z}[\chi]$, and so the computation of
$\tilde{V}$ is well-defined over $\mathbb{F}_{q}$. ∎

The resultant matrix $\tilde{V}$ will clearly contain the projection of
$\mathcal{S}_{1}(N,\chi)$ onto $\mathbb{F}_{q}$ but may also contain ethereal
forms, and modular functions. These are subsequently removed by
diagonalisation and lifting. We will show that forms in
$\mathcal{S}_{1}^{\text{new}}(N,\chi)$ are recovered by this process. We need
the following lemma:

######  Lemma 6.

If $\alpha$ and $\beta$ are roots of unity, with
$\operatorname{Ord}\left(\frac{\alpha}{\beta}\right)\mid a$ and
$\operatorname{Ord}(\alpha\beta)\mid b$, then $\operatorname{Ord}(\alpha)\mid
2\operatorname{LCM}(a,b)$.

###### Proof.

By assumption, $\alpha^{a}=\beta^{a}$, and thus
$(\alpha\beta)^{a}=\alpha^{2a}$. Let $h=2\operatorname{LCM}(a,b)$, then
$2a\mid h\implies\alpha^{h}=(\alpha\beta)^{\frac{h}{2}}$ and as
$b\mid\frac{h}{2}$ we conclude $\alpha^{h}=1$. ∎

Using this, we restrict the possible orders of Satake parameters of weight 1
newforms.

######  Lemma 7.

Fix $f$ a weight 1 eigenform of level $N$ and character $\chi$. Let $D$ be the
LCM of all orders of projective images of Artin representations associated
with eigenbasis elements for $\mathcal{S}_{1}^{\text{dihedral}}(M,\chi)$ where
$M$ ranges across all values satisfying $\mathfrak{f}(\chi)\mid M\mid N$, and
let $H=2\operatorname{LCM}(60,\phi(N),D)$. Any Satake paramter of $f$ has
order dividing $H$.

###### Proof.

The action of $\text{GL}_{2}(\mathbb{C})$ by conjugation on $2\times 2$
matrices with trace 0 gives an isomorphism
$\text{PGL}_{2}(\mathbb{C})\cong\text{SO}_{3}(\mathbb{C})$. Thus, the
projection of a 2-dimensional Artin representation gives a 3-dimensional
representation of the projective image.

For a matrix $M_{2}\in\text{GL}_{2}(\mathbb{C})$ the characteristic polynomial
is

$x^{2}-\text{tr}(M_{2})x+\det(M_{2}),$ (32)

and the characteristic polynomial of the associated matrix $M_{3}\in
SO_{3}(\mathbb{C})$ is:

$x^{3}+\left(1-\frac{\text{tr}(M_{2})^{2}}{\det(M_{2})}\right)(x^{2}-x)-1.$
(33)

Supposing the roots of (32) are $\alpha$ and $\beta$ then the roots of (33)
are $\frac{\alpha}{\beta},\frac{\beta}{\alpha}$ and 1.

Let $\rho$ be a two-dimensional Artin representation with projective image
$D_{n}$, and fix a prime $p\nmid N$. The above isomorphism associates with
$\rho$ a 3-dimensional representation of $D_{n}$ with character value
$\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+1$ at $\overline{\rho}(p)$. The
representation theory of dihedral groups gives finite possibilities for this
character value, and so restricts the possible order of the root of unity
$\frac{\alpha}{\beta}$. If $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+1=3$
then $\frac{\alpha}{\beta}=1$. If
$\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+1=1$ then
$\operatorname{Ord}\left(\frac{\alpha}{\beta}\right)=4$, and if
$\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+1=1+2\cos(2\pi ik/n)$ for some
$k\in\mathbb{N}$ then
$\operatorname{Ord}\left(\frac{\alpha}{\beta}\right)=\frac{n}{(n,k)}$. In all
cases, we have
$\operatorname{Ord}\left(\frac{\alpha}{\beta}\right)\mid\operatorname{LCM}(4,n)$.

Equivalent analysis of the groups $S_{4},A_{4}$ and $A_{5}$ reveals that in
the case of exotic representations, we must have
$\operatorname{Ord}\left(\frac{\alpha}{\beta}\right)\mid 60$. Together, with
$D$ defined as in the lemma, we conclude that for any eigenform
$f\in\mathcal{S}_{1}^{\text{new}}(N,\chi)$ with Satake parameters $\alpha$ and
$\beta$ at $p\nmid N$ we must have
$\operatorname{Ord}\left(\frac{\alpha}{\beta}\right)\mid\operatorname{LCM}(60,D)$.
Apply Lemma 6 to:

$\operatorname{Ord}\left(\frac{\alpha}{\beta}\right)\mid\operatorname{LCM}(60,D),\;\;\;\operatorname{Ord}(\alpha\beta)\mid\operatorname{Ord}(\chi).$
(34)

This gives $\operatorname{Ord}(\alpha)\mid
2\operatorname{LCM}(60,\operatorname{Ord}(\chi),D)\mid H$. For $p\mid N$ there
is at most one Satake parameter, whose order must divide the order of the
projective image, and therefore $H$. ∎

Consequently, if $d$ is the dimension of
$\mathcal{S}_{1}^{\text{new}}(N,\chi)$, there exists a linear transformation
of $\tilde{V}$ such that the final $d$ columns are the projection onto
$\mathbb{F}_{q}$ of an eigenbasis of $\mathcal{S}_{1}^{\text{new}}(N,\chi)$.
The next step is to diagonalise $\tilde{V}$ to acquire such a matrix.

Diagonalisation is performed as follows. First, we find a linear
transformation of $\tilde{V}$ such that the leftmost columns form a basis for
$\mathcal{S}_{1}^{\text{old}}(N,\chi)$. The action of any Hecke operator $T$
on this matrix is of the form:

$\left(\begin{array}[]{c|c}A^{\text{old}}&*\\\ \hline\cr 0&A^{\text{new}}\\\
\end{array}\right),$ (35)

where $A^{\text{old}}$ is a square matrix representing the action of $T$ on
$\mathcal{S}_{1}^{\text{old}}(N,\chi)$. Given that $\tilde{V}$ has been
computed up to at least the Sturm bound, there exists some linear combination
of Hecke operators such that the eigenvalues of $A^{\text{new}}$ are unique
and distinct from the eigenvalues of $A^{\text{old}}$. These eigenvalues are
found by computing the roots of the characteristic polynomial of
$A^{\text{new}}$ over $\mathbb{F}_{q}$. This polynomial may not split
completely, indicating ethereal forms and or modular functions whose Fourier
expansions lie in some extension of $\mathbb{F}_{q}$.

By taking just the linear factors of the characteristic polynomial, removing
oldform contribution to the resulting eigenvectors, and normalising so that
$a_{n}=1$, we construct a set of basis elements which contains the projection
of an eigenbasis of $\mathcal{S}_{1}^{\text{new}}(N,\chi)$ onto
$\mathbb{F}_{q}$. In our computation up to level 10,000, the resulting space
was in fact always equal to the projection of an eigenbasis of
$\mathcal{S}_{1}^{\text{new}}(N,\chi)$, but in theory it could still contain
ethereal forms and modular functions.

The Satake parameters are subsequently found by factorising
$x^{2}-a_{p}x+\chi(p)$ for all primes $p$ (as in the diagonalisation step, any
form for which we cannot compute all Satake parameters must be either an
ethereal form or modular function, so can be discarded). These parameters are
lifted back to $\mathbb{Z}[\zeta_{H}]$ by sending each element to the root of
unity in its preimage in the original projection mapping $\sigma$. Hecke
relations are then used to define all $a_{n}$ for all composite $n$ up to the
Sturm bound.

######  Lemma 8.

Diagonalising and lifting $\tilde{V}$ results in a matrix containing an
eigenbasis for $\mathcal{S}_{1}^{\text{new}}(N,\chi)$.

###### Proof.

Let $f$ be a form in the eigenbasis for
$\mathcal{S}_{1}^{\text{new}}(N,\chi)$. From Lemma 7, we see that any Satake
parameter $\alpha_{p}$ is in $\mathbb{Z}[\zeta_{H}]$, and so for any $p\mid N$
we have $a_{p}\in\mathbb{Z}[\zeta_{H}]$. For $p\nmid N$ we have
$a_{p}=\alpha_{p}+\chi(p)\overline{\alpha_{p}}$ and so again
$a_{p}\in\mathbb{Z}[\zeta_{H}]$. From Hecke relations, this is true for all
$a_{n}$.

Consequently, the eigenvalues of projections of forms in
$\mathcal{S}_{1}^{\text{new}}(N,\chi)$ onto $\mathbb{F}_{q}$ will be present
in the diagonalisation of $\tilde{V}$. The restriction of the original
projection mapping $\sigma$ to the cyclic group generated by $\zeta_{H}$ is
injective, and so recovery of the Fourier coefficients by lifting follows from
Lemma 7. ∎

For any $f\in\mathcal{S}_{1}^{\text{new}}(N,\chi)$, we have thus computed the
projection of $f$ onto $\mathbb{F}_{q}$, and then recovered the original form
via Satake parameters. However, it is still not guaranteed that all of the
resulting functions are truncations of forms in
$\mathcal{S}_{1}^{\text{new}}(N,\chi)$. To verify the computation, we proceed
as follows. Fix an Eisenstein series
$E\in\mathcal{E}_{1}^{\text{basis}}(N,\chi)$, then:

$f\in\mathcal{S}_{1}^{\text{new}}(N,\chi)\iff
f\overline{E}\in\mathcal{S}_{2}(N,\mathbbm{1}).$ (36)

This is because $f$ is (by construction) Hecke stable, and (by Lemma 4) not an
Eisenstein series, and so the characteristic 0 case of the Hecke stability
method holds. This completes the proof of Theorem 1.

##  4\. Computation up to level $10,000$

The computation up to the $n$-th coefficient of $d$ basis elements of a $d$
dimensional cusp form space from its trace form is
$O\left((nd)^{\frac{3}{2}}\right)$, as shown in [10]. In order to perform the
stabilisation step of our algorithm, we require $O(dp)$ coefficients of a
basis for $\mathcal{S}_{2}(N,\mathbbm{1})$ where $p$ is the least prime not
dividing $N$. This is most efficiently performed by decomposing the space into
the smallest possible subspaces; twists from twist-minimal spaces. For
example, when computing the weight 1 forms at level $175$, we require $30$
coefficients of $15$ basis elements of $\mathcal{S}_{2}(175,\mathbbm{1})$.
This would involve computation up to the $330$-th coefficient of the trace
form of $\mathcal{S}_{2}^{\text{new}}(175,\mathbbm{1})$, but only up to the
$120$-th coefficient of the trace form of
$\mathcal{S}_{2}^{\min}(175,\mathbbm{1})$.

This computation gives us a methods of estimating the computation time up to a
given level. By [20, Theorem 8], the asymptotic expression for the dimension
of $\mathcal{S}_{2}(N,\mathbbm{1})$ is $O(N)$. Along with the Sturm bound,
this gives the complexity of computing the initial matrix for
$\mathcal{S}_{2}(N,\mathbbm{1})$ as $O(N^{3})$. Consequently, computing up to
level $N_{\max}$ has complexity $O(N_{\max}^{4})$. The time taken in our
computations up to various levels suggests the estimate given by a degree 4
polynomial is reasonable. The following table is the time required for 192
cores to compute up to level $N_{\max}$ along with predicted time from fitting
a degree 4 polynomial:

$N_{\max}$ | Minutes to compute | Predicted minutes to compute
---|---|---
2,000 | 1 | 5
3,000 | 7 | -3
4,000 | 19 | 18
5,000 | 53 | 64
6,000 | 130 | 136
7,000 | 245 | 238
8,000 | 398 | 377
9,000 | 541 | 567
10,000 | 832 | 822
15,000 | - | 3,905
20,000 | - | 13,251
30,000 | - | 78,708
40,000 | - | 277,629

Dashes indicate that the level was not computed up to. Note that the
anticipated computation time for $N_{\max}=20,000$ makes such a computation
seem feasible. In practice, the main barrier we faced was not computation time
but the memory requirements for finding a left pseudo-inverse of the weight 2
basis matrix over $\mathbb{Q}$. Such a computation is required for verifying
$f\overline{E}\in\mathcal{S}_{2}(N,\mathbbm{1})$ in the final step of the
algorithm, but inverting matrices of approximate size $2,000\times 2,000$ has
a large memory overhead, such that even with 1TB of RAM we opted to perform
these inversions one at a time past level $9,000$.

The final verification step involves a computationally expensive
multiplication over a number field. The improvement in our approach comes from
the fact that this only needs to be performed on a small number of elements
(roughly the size of the exotic space). In contrast, methods which only use
the characteristic $p$ computation as an upper bound would have to perform
number field computations for all elements in weight 2. For example, at level
$N=124$ we multiply two elements in
$\mathcal{S}_{1}^{\text{new}}(N,\chi_{124}(67,\cdot))$111Here,
$\chi_{124}(67,\cdot)$ refers to the Conrey label of the character, defined in
[2] rather than dividing fourteen elements in
$\mathcal{S}_{2}(N,\mathbbm{1})$.

To compute all spaces up to level $10,000$, we could simply use Theorem 1
directly on every level and character up to this bound, but such a computation
is improved by the following result reducing the scope of the search.

######  Lemma 9.

Let $p\mid N$ be any prime satisfying
$\nu_{p}(N)\in\\{1,\nu_{p}(\mathfrak{f}(\chi))\\}$. If $\chi_{p}=\mathbbm{1}$
then $\mathcal{S}_{1}^{\text{new}}(N,\chi)=0$, and if
$\operatorname{Ord}(\chi_{p})>5$ then
$\mathcal{S}_{1}^{\text{exotic}}(N,\chi)=0$. Further, if $p$ and $q$ are two
such primes, with $\operatorname{Ord}(\chi_{p}\chi_{q})=20$, then
$\mathcal{S}_{1}^{\text{exotic}}(N,\chi)=0$.

###### Proof.

The results on $\mathcal{S}_{1}^{\text{exotic}}$ are given in [6]. The
additional result on all of $\mathcal{S}^{\text{new}}$ comes from the remarks
after (20). ∎

In general, for a level $N$ and character $\chi$, we only need to compute
$\mathcal{S}_{1}^{\text{new}}(N,\chi)$ if it is possible that
$\mathcal{S}_{1}^{\text{exotic}}(N,\chi)$ is non-zero, or if there exists some
multiple $M$ of $N$ which we wish to compute where it is possible that
$\mathcal{S}_{1}^{\text{exotic}}(M,\chi)$ is non-zero, such that we will
require lifts of the elements of $\mathcal{S}_{1}^{\text{dihedral}}(N,\chi)$
as oldforms. This requirement drives Definition 2.4 and Lemma 2 follows
trivially.

## References

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∎

11institutetext: M. YOUNUS BHAT 22institutetext: Department of Mathematical
Sciences, Islamic University of Science and Technology Awantipora, Pulwama,
Jammu and Kashmir 192122, India. 22email<EMAIL_ADDRESS>33institutetext:
AAMIR H. DAR 44institutetext: Department of Mathematical Sciences, Islamic
University of Science and Technology Awantipora, Pulwama, Jammu and Kashmir
192122, India.

# Convolution and Correlation Theorems for Wigner-Ville Distribution
Associated with the Quaternion Offset Linear Canonical Transform

Convolution and Correlation Theorems for WVD Associated with the QOLCT

M. YOUNUS BHAT  AAMIR H. DAR

(Received: date / Accepted: date)

###### Abstract

The quaternion offset linear canonical transform(QOLCT) has gained much
popularity in recent years because of its applications in many areas,
including color image and signal processing. At the same time the applications
of Wigner-Ville distribution (WVD) in signal analysis and image processing can
not be excluded. In this paper we investigate the Winger-Ville Distribution
associated with quaternion offset linear canonical transform (WVD-QOLCT).
Firstly, we propose the definition of the WVD-QOLCT, and then several
important properties of newly defined WVD-QOLCT, such as nonlinearity,
bounded, reconstruction formula, orthogonality relation and Plancherel formula
are derived. Secondly a novel canonical convolution operator and a related
correlation operator for WVD-QOLCT are proposed. Moreover, based on the
proposed operators, the corresponding generalized convolution, correlation
theorems are studied.We also show that the convolution and correlation
theorems of the QWVD and WVD-QLCT can be looked as a special case of our
achieved results.

###### Keywords:

Quaternion algebra Offset linear canonical transform Quaternion offset linear
canonical transform Wigner-Ville distribution Convolution
CorrelationModulation

###### MSC:

11R52 42C40 42C3043A30.

##  1 Introduction

In the time-frequency signal analysis the classical Wigner-Ville distribution
(WVD) or Wigner- Ville transform (WVT) has an important role to play. Eugene
Wigner introduced the concept WVD while making his calculation of the quantum
corrections. later on it was J. Ville who derived it independently as a
quadratic representation of the local time-frequency energy of a signal in
1948. Many important properties of WVT has been studied by many authors. On
replacing the kernel of the classical Fourier transform (FT) with the kernel
of the LCT in the WVD domain, this transform can be extended to the domain of
linear canonical transform 3 -6 , 13 -19 .

On the other hand the quaternion Fourier transform (QFT) is of the interest in
the present era. Many important properties like shift, modulation,
convolution, correlation, differentiation, energy conservation, uncertainty
principle of QFT have been found. Many generalized transforms are closely
related to the QFTs, for example, the quaternion wavelet transform, fractional
quaternion Fourier transform, quaternion linear canonical transform, and
quaternionic windowed Fourier transform. Based on the QFTs, one also may
extend the WVD to the quaternion algebra while enjoying similar properties as
in the classical case. Many authors generalized the classical WVD to
quaternion algebra, which they called as the quaternion Wigner-Ville
distribution (QWVD). For more details we refer to 1 , 2 , 7 -12 .

The linear canonical transform (LCT) with four parameters $(a,b,c,d)$ has been
generalized to a six parameter transform $(a,b,c,d,u_{0},w_{0})$ known as
offset linear canonical transform (OLCT). Due to the time shifting $u_{0}$ and
frequency modulation parameters, the OLCT has gained more flexibility over
classical LCT. Hence has found wide applications in image and signal
processing. On the other side the convolution has some applications in various
areas of Mathematics like linear algebra, numerical analysis and signal
processing. Where as Correlation like convolution is an another important tool
n signal processing, optics and detection applications. In the domains of LCT,
WVD and OLCT the convolution and correlation operations have been studied 7
-10 .

The quaternion offset linear canonical transform
(QOLCT) has gained much popularity in recent years because of its applications
in many areas, including colour image and signal processing. At the same time
the applications of Wigner-Ville distribution (WVD) in signal analysis and
image processing can not be excluded. Motivated by QOLCT and WVD, we in this
paper we investigate the Winger-Ville Distribution associated with quaternion
offset linear canonical transform (WVD-QOLCT). Firstly, we propose the
definition of the WVD-QOLCT, and then several important properties of newly
defined WVD-QOLCT, such as nonlinearity, bounded, reconstruction formula,
orthogonality relation and Plancherel formula are derived. Secondly a novel
canonical convolution operator and a related correlation operator for WVD-
QOLCT are proposed. Moreover, based on the proposed operators, the
corresponding generalized convolution, correlation theorems are studied.We
also show that the convolution and correlation theorems of the QWVD and WVD-
QLCT can be looked as a special case of our achieved results.

The paper is organised as follows. In Section 2, we provide the definition of
Wigner-Ville distribution associated with the quaternionic offset linear
canonical transform (WVD-QOLCT). Then we will investigate several basic
properties of the WVD-QOLCT which are important for signal representation in
signal processing. In Section 3 we first define the convolution and
correlation for the QOLCT. We then establish the new convolution and
correlation for the WVD-QOLCT.We also show that the convolution theorems of
the QWVD and WVD-QLCT can be looked as a special case of our achieved results
I

##  2 Winger-ville Distribution associated with Quaternion Offset Linear
Canonical Transform(WVD-QOLCT)

Since in practice most natural signals are non-stationary. In order to study a
non-stationary signals the Wigner-Ville distribution has become a suite tool
for the analysis of the non stationary signals. In this section,we are going
to give the definition of Wigner-Ville distribution associated with the
quaternionic offset linear canonical transform (WVD-QOLCT),then we will
investigate several basic properties of the WVD-QOLCT which are important for
signal representation in signal processing.

###### Definition 1

Let $A_{i}=\left[\begin{array}[]{cccc}a_{i}&b_{i}&|&r_{i}\\\
c_{s}&d_{i}&|&s_{i}\\\ \end{array}\right]$, be a matrix parameter such that
$a_{s}$, $b_{i}$, $c_{i}$, $d_{i}$, $r_{i}$, $s_{i}\in\mathbf{R}$ and
$a_{i}d_{i}-b_{i}c_{i}=1,$ for $i=1,2.$ The Wigner-Ville distribution
associated with the two-sided quaternionic offset linear canonical transform
(WVD-QOLCT) of signals $f,g\in L^{2}(\mathbf{R}^{2},\mathbf{H})$, is given by

${\mathcal{W}}_{f,g}^{A_{1},A_{2}}(t,u)=\left\\{\begin{array}[]{ll}\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}f(t+\frac{n}{2})\overline{g(t-\frac{n}{2})}\\\
\ \quad K^{j}_{A_{2}}(n_{2},u_{2})dn,\\\ \ \quad b_{1},b_{2}\neq 0,\\\ \
\sqrt{d_{1}}{e}^{i(\frac{c_{1}d_{1}}{2}{\left(u_{1}-r_{1}\right)}^{2}+{\
u}_{1}r_{1})}\\\ \ \quad
f(t_{1}+\frac{d_{1}{(u}_{1}-r_{1})}{2},t_{2}+\frac{n_{2}}{2})\\\ \
\quad\overline{g}(t_{1}-\frac{d_{1}{(u}_{1}-r_{1})}{2},t_{2}-\frac{n_{2}}{2})\\\
\ K^{j}_{A_{2}}\left(n_{2},u_{2}\right),\\\ \ \quad b_{1}=0,b_{2}\neq 0;\\\ \
\sqrt{d_{2}}{K^{i}_{A_{1}}\left(n_{1},u_{1}\right)}\\\ \ \quad
f(t_{1}+\frac{n_{1}}{2},t_{2}+\frac{d_{2}{(u}_{2}-r_{2})}{2})\\\ \
\quad\overline{g}(t_{1}-\frac{n_{1}}{2},t_{2}-\frac{d_{2}{(u}_{2}-r_{2})}{2})\\\
\ \quad e^{j(\frac{c_{2}d_{2}}{2}{(u_{2}-r_{2})}^{2}+{\ u}_{2}{r}_{2)}},\\\ \
\quad b_{1}\neq 0,b_{2}=0;\\\ \
\sqrt{d_{1}d_{2}}e^{i(\frac{c_{1}d_{1}}{2}{\left(u_{1}-r_{1}\right)}^{2}+{u}_{1}r_{1})}\\\
\ \quad
f(t_{1}+\frac{d_{1}({u}_{1}-{r}_{1}}{2}),t_{2}+\frac{d_{2}(u_{2}-r_{2}}{2}))\\\
\
\quad\overline{g}(t_{1}-\frac{d_{1}({u}_{1}-{r}_{1}}{2}),t_{2}-\frac{d_{2}(u_{2}-r_{2}}{2}))\\\
\ \quad e^{j(\frac{c_{2}d_{2}}{2}{(u_{2}-r_{2})}^{2}+{u}_{2}{r}_{2})},\\\ \
\quad b_{1}=b_{2}=0.\end{array}\right.$ (1)

where $t=(t_{1},t_{2}),u=(u_{1},u_{2}),n=(n_{1},n_{2})$ and
$K_{A_{1}}^{i}(n_{1},u_{1})$ and $K_{A_{2}}^{j}(n_{2},u_{2})$ are the
quaternion kernels.

###### Note 1

If $f=g$ then $\mathcal{W}_{f,f}^{A_{1},A_{2}}(t,u)$ we call it the Auto WVD-
QOLCT.Otherwise is is called Cross WVD-QOLCT

Without loss of generality we will deal with the case $b_{i}\neq 0,\,i=1,2,$
as in other cases proposed transform reduces to a chrip multiplications.Thus
for any $f,g\in L^{2}(\mathbf{R}^{2},\mathbf{H})$ we have

$\displaystyle\mathcal{W}_{f,g}^{A_{1},A_{2}}(t,u)$ $\displaystyle=$
$\displaystyle\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}f(t+\frac{n}{2})\overline{g(t-\frac{n}{2})}K^{j}_{A_{2}}(n_{2},u_{2})dn$
$\displaystyle=$
$\displaystyle\mathcal{O}^{i,j}_{A_{1},A_{2}}\left\\{f\left(t+\frac{n}{2}\right)\overline{g\left(t-\frac{n}{2}\right)}\right\\}$
$\displaystyle=$
$\displaystyle\mathcal{O}^{i,j}_{A_{1},A_{2}}\\{h_{f,g}(t,n)\\}.$ (2)

Where
$h_{f,g}(t,n)=f\left(t+\frac{n}{2}\right)\overline{g\left(t-\frac{n}{2}\right)}$
is known as quaternion correlation product. Applying the inverse QOLCT to (2),
we get

$\\{h_{f,g}(t,n)\\}=\\{\mathcal{O}_{A_{1},A_{2}}^{i,j}\\}^{-1}\\{\mathcal{W}_{f,g}^{A_{1},A_{2}}(t,u)\\}$

which implies

$\displaystyle
f\left(t+\frac{n}{2}\right)\overline{g\left(t-\frac{n}{2}\right)}$
$\displaystyle=$
$\displaystyle\\{\mathcal{O}_{A_{1},A_{2}}^{i,j}\\}^{-1}\\{\mathcal{W}_{f,g}^{A_{1},A_{2}}(t,u)\\}$
$\displaystyle=$
$\displaystyle\int_{\mathbf{R}^{2}}K_{A_{1}}^{-i}(t_{1},u_{1})\mathcal{W}_{f,g}^{A_{1},A_{2}}(t,u)$
$\displaystyle K_{A_{2}}^{-j}(t_{2},u_{2})dw.$ (3)

Now, we discuss several basic properties of the WVD-QOLCT given by (1). These
properties play important roles in signal representation.

###### Theorem 2.1

(Boundedness) Let $f,g\in L^{2}(\mathbf{R}^{2},\mathbf{H}).$ Then

$\big{|}\mathcal{W}_{f,g}^{A_{1},A_{2}}(t,u)\big{|}\leq\frac{2}{\pi\sqrt{b_{1}b_{2}}}\|f\|_{L^{2}\mathbf{(}{R}^{2},H)}\|g\|_{L^{2}\mathbf{(}{R}^{2},H)}$
(4)

###### Proof

By the virtue of Cauchy-Schwarz inequality in quaternion domain, we have

$\displaystyle|\mathcal{W}_{f,g}^{A_{1},A_{2}}(t,u)|^{2}$
$\displaystyle=\left|\int_{\mathbf{R}^{2}}K_{A_{1}}^{i}(n_{1},u_{1})\,f\left(t+\frac{n}{2}\right)\overline{g\left(t-\frac{n}{2}\right)}\,K_{A_{2}}^{j}(n_{2},u_{2})dn\right|^{2}$
$\displaystyle\leq\left(\int_{\mathbf{R}^{2}}\left|K_{A_{1}}^{\mathbf{i}}(n_{1},u_{1})f\left(t+\frac{n}{2}\right)\overline{g\left(t-\frac{n}{2}\right)}K_{A_{2}}^{\mathbf{j}}(n_{2},u_{2})\right|\rm{d}n\right)^{2}$
$\displaystyle=\left(\frac{1}{\sqrt{4\pi^{2}|b_{1}b_{2}|}}\int_{\mathbf{R}^{2}}\left|f\left(t+\frac{n}{2}\right)\overline{g\left(t-\frac{n}{2}\right)}\right|\rm{d}n\right)^{2}$
$\displaystyle\leq\frac{1}{4\pi^{2}|b_{1}b_{2}|}\left(\int_{\mathbf{R}^{2}}\left|f\left(t+\frac{n}{2}\right)\right|^{2}\rm{d}n\right)\left(\int_{\mathbf{R}^{2}}\left|\overline{g\left(t-\frac{n}{2}\right)}\right|^{2}\rm{d}s\right)$
$\displaystyle=\frac{1}{4\pi^{2}|b_{1}b_{2}|}\left(4\int_{\mathbf{R}^{2}}\left|f(w)\right|^{2}\rm{d}w\right)\left(4\int_{\mathbf{R}^{2}}\left|\overline{g(y)}\right|^{2}\rm{d}y\right)$
$\displaystyle=\frac{4}{\pi^{2}|b_{1}b_{2}|}\|f\|_{L^{2}\mathbf{(}{R}^{2},H)}^{2}\|g\|_{L^{2}\mathbf{(}{R}^{2},H)}^{2}$

where applying the change of variables $w=t+\frac{n}{2}$ and $y=t-\frac{n}{2}$
in the last second step. Then we have

$\displaystyle|\mathcal{W}_{f,g}^{A_{1},A_{2}}(t,u)|\leq\frac{2}{\pi\sqrt{|b_{1}b_{2}|}}\|f\|_{L^{2}\mathbf{(}{R}^{2},H)}\|g\|_{L^{2}\mathbf{(}{R}^{2},H)}$

which completes the proof of Theorem.

###### Theorem 2.2

(Nonlinearity) Let $f$ and $g$ be two quaternion functions in
$L^{2}(\mathbf{R}^{2},\mathbf{H}).$ Then

$\mathcal{W}_{f+g}^{A_{1},A_{2}}=\mathcal{W}_{f,f}^{A_{1},A_{2}}+\mathcal{W}_{f,g}^{A_{1},A_{2}}+\mathcal{W}_{g,f}^{A_{1},A_{2}}+\mathcal{W}_{g,g}^{A_{1},A_{2}}$
(5)

###### Proof

By definition 1 we have

$\displaystyle\mathcal{W}_{f+g}^{A_{1},A_{2}}(t,u)$
$\displaystyle=\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}\left[f(t+\frac{n}{2})+{g}(t+\frac{n}{2})\right]$
$\displaystyle\overline{\left[f(t-\frac{n}{2})+{g}(t-\frac{n}{2})\right]}K^{j}_{A_{2}}(n_{2},u_{2})dn$
$\displaystyle=\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}\left[f(t+\frac{n}{2})\overline{f(t-\frac{n}{2})}\right.$
$\displaystyle+f(t+\frac{n}{2})\overline{g(t-\frac{n}{2})}$
$\displaystyle+\left.g(t+\frac{n}{2})\overline{f(t-\frac{n}{2})}+g(t+\frac{n}{2})\overline{g(t-\frac{n}{2})}\right]K^{j}_{A_{2}}(n_{2},u_{2})dn$
$\displaystyle=\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}f\left(t+\frac{n}{2}\right)\overline{f\left(t-\frac{n}{2}\right)}K^{j}_{A_{2}}(n_{2},u_{2})dn$
$\displaystyle\;+\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}f\left(t+\frac{n}{2}\right)\overline{g\left(t-\frac{n}{2}\right)}K^{j}_{A_{2}}(n_{2},u_{2})dn$
$\displaystyle\;+\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}g\left(t+\frac{n}{2}\right)\overline{f\left(t-\frac{n}{2}\right)}K^{j}_{A_{2}}(n_{2},u_{2})dn$
$\displaystyle\;+\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}g\left(t+\frac{n}{2}\right)\overline{g\left(t-\frac{n}{2}\right)}K^{j}_{A_{2}}(n_{2},u_{2})dn$
$\displaystyle=\mathcal{W}_{f,f}^{A_{1},A_{2}}+\mathcal{W}_{f,g}^{A_{1},A_{2}}+\mathcal{W}_{g,f}^{A_{1},A_{2}}+\mathcal{W}_{g,g}^{A_{1},A_{2}}$

which completes the proof of Theorem.

Note the properties like Shift,Modulation,Dilation are similar to the
classical QOLCT so we avoided them.

###### Theorem 2.3

(Reconstruction formula). For $f,g\in L^{2}(\mathbf{R}^{2},\mathbf{H})$ where
$g$ does not vanish at $0$ .We get the following inversion formula of the WVD-
QOLCT:

$\displaystyle f(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\overline{g(0)}}\int_{\mathbf{R}^{2}}K^{-i}_{A_{1}}(u_{1},n_{1})\mathcal{W}_{f,g}^{A_{1},A_{2}}\left(\frac{t}{2},u\right)$
$\displaystyle K^{-j}_{A_{2}}(u_{2},n_{2})du$ (6)

###### Proof

By (3), we have

$\displaystyle\\{h_{f,g}(t,n)\\}$
$\displaystyle=\\{\mathcal{O}_{A_{1},A_{2}}^{i,j}\\}^{-1}\\{\mathcal{W}_{f,g}^{A_{1},A_{2}}(t,u)\\}$

which implies

$\displaystyle
f\left(t+\frac{n}{2}\right)\overline{g\left(t-\frac{n}{2}\right)}$
$\displaystyle=\int_{\mathbf{R}^{2}}K_{A_{1}}^{-i}(t_{1},u_{1})\mathcal{W}_{f,g}^{A_{1},A_{2}}(t,u)K_{A_{2}}^{-j}(t_{2},u_{2})dw,$

Now let $t=\frac{n}{2}$ and taking change of variable $w=2t$, we get

$f(w)=\frac{1}{\overline{g(0)}}\int_{\mathbf{R}^{2}}K^{-i}_{A_{1}}(u_{1},n_{1})\mathcal{W}_{f,g}^{A_{1},A_{2}}\left(\frac{w}{2},u\right)K^{-j}_{A_{2}}(u_{2},n_{2})du$

which completes the proof of Theorem.

###### Theorem 2.4

(Orthogonality relation). If $f_{1},f_{2},g_{1},g_{2}\in
L^{2}(\mathbf{R}^{2},\mathbf{H})$ are quaternion-valued signals.Then

$\left\langle\mathcal{W}^{A_{1},A_{2}}_{f_{1},g_{1}}(t,u),{\mathcal{W}^{A_{1},A_{2}}_{f_{2},g_{2}}(t,u)}\right\rangle=\left[\left\langle
f_{1},f_{2}\right\rangle\left\langle
g_{2},g_{1}\right\rangle\right]_{\mathbf{H}}$ (7)

###### Proof

By the definition of Winger-ville distribution associated with quaternion OLCT
and innear product relation we have

$\displaystyle\langle
W^{A_{1},A_{2}}_{f_{1},g_{1}}(t,u),{\mathcal{W}^{A_{1},A_{2}}_{f_{2},g_{2}}(t,u)}\rangle$
$\displaystyle=\int_{\mathbf{R}^{4}}\left[\mathcal{W}^{A_{1},A_{2}}_{f_{1},g_{1}}(t,u)\overline{{\mathcal{W}^{A_{1},A_{2}}_{f_{2},g_{2}}(t,u)}}\right]_{\mathbf{H}}dudt$
$\displaystyle=\int_{\mathbf{R}^{4}}\left[\mathcal{W}^{A_{1},A_{2}}_{f_{1},g_{1}}(t,u)\right.$
$\displaystyle\left.\overline{\int_{\mathbf{R}^{2}}K_{A_{1}}^{i}(n_{1},u_{1})f_{2}\left(t+\frac{n}{2}\right)\overline{g_{2}\left(t-\frac{n}{2}\right)}}\right.$
$\displaystyle\left.\overline{K_{A_{2}}^{j}(n_{2},u_{2})dn}\right]_{\mathbf{H}}dudt$
$\displaystyle=\int_{\mathbf{R}^{6}}\left[\mathcal{W}^{A_{1},A_{2}}_{f_{1},g_{1}}(t,u)K_{A_{2}}^{-j}(n_{2},u_{2})g_{2}\left(t-\frac{n}{2}\right)\right.$
$\displaystyle\left.\overline{f_{2}\left(t+\frac{n}{2}\right)}K_{A_{1}}^{-i}(n_{1},u_{1})\right]_{\mathbf{H}}dudtdn$
$\displaystyle=\int_{\mathbf{R}^{6}}\left[K_{A_{1}}^{-i}(n_{1},u_{1})\mathcal{W}^{A_{1},A_{2}}_{f_{1}g_{1}}(t,u)K_{A_{2}}^{-j}(n_{2},u_{2})\right.$
$\displaystyle\left.g_{2}\left(t-\frac{n}{2}\right)\overline{f_{2}\left(t+\frac{n}{2}\right)}\right]_{\mathbf{H}}dudtdn$
$\displaystyle=\int_{\mathbf{R}^{4}}\left[\int_{\mathbf{R}^{2}}K_{A_{1}}^{-i}(n_{1},u_{1})\mathcal{W}^{A_{1},A_{2}}_{f_{1},g_{1}}(t,u)K_{A_{2}}^{-j}(n_{2},u_{2})du\right.$
$\displaystyle\left.g_{2}\left(t-\frac{n}{2}\right)\overline{f_{2}\left(t+\frac{n}{2}\right)}\right]_{\mathbf{H}}dtdn$
(8)

Because

$\displaystyle\overline{K_{A_{1}}^{i}(n_{1},u_{1})}=K_{A_{1}}^{-i}(u_{1},n_{1})=K_{A_{1}^{-1}}^{i}(u_{1},n_{1})$
$\displaystyle\overline{K_{A_{2}}^{j}(n_{2},u_{2})}=K_{A_{2}}^{-j}(n_{2},u_{2})=K_{A_{2}^{-1}}^{j}(u_{2},n_{2})$

Now by using (3) in (8), we have

$\displaystyle\left\langle\mathcal{W}^{A_{1},A_{2}}_{f_{1},g_{1}}(t,u),{\mathcal{W}^{A_{1},A_{2}}_{f_{2},g_{2}}(t,u)}\right\rangle$
$\displaystyle=\int_{\mathbf{R}^{4}}\left[\int_{\mathbf{R}^{2}}K_{A_{1}^{-1}}^{i}(u_{1},n_{1})\mathcal{W}^{A_{1},A_{2}}_{f_{1},g_{1}}(t,u)K_{A_{2}^{-1}}^{j}(u_{2},n_{2})du\right.$
$\displaystyle\left.g_{2}\left(t-\frac{n}{2}\right)\overline{f_{2}\left(t+\frac{n}{2}\right)}\right]_{\mathbf{H}}dtdn$
$\displaystyle=\int_{\mathbf{R}^{4}}\left[f_{1}\left(t+\frac{n}{2}\right)\overline{g_{1}\left(t-\frac{n}{2}\right)}g_{2}\left(t-\frac{n}{2}\right)\overline{f_{2}\left(t+\frac{n}{2}\right)}\right]_{\mathbf{H}}dtdn$

Using the change of variables $t+\frac{n}{2}=\omega$,and $t-\frac{n}{2}=\xi$
the equation becomes

$\displaystyle\langle\mathcal{W}^{A_{1},A_{2}}_{f_{1},g_{1}}(t,u),\mathcal{W}^{A_{1},A_{2}}_{f_{2},g_{2}}(t,u)\rangle$
$\displaystyle=\int_{\mathbf{R}^{4}}\left[f_{1}(\omega)\overline{g_{1}(\xi)}g_{2}(\xi)\overline{f_{2}(\omega)}\right]_{\mathbf{H}}d\omega
d\xi$
$\displaystyle=\left[\int_{\mathbf{R}^{2}}f_{1}(\omega)\overline{f_{2}(\omega)}d\omega\int_{\mathbf{R}^{2}}g_{2}(\xi)\overline{g_{1}(\xi)}d\xi\right]_{\mathbf{H}}$
$\displaystyle=[\langle f_{1},f_{2}\rangle\langle
g_{2},g_{1}\rangle]_{\mathbf{H}}$

which completes the proof theorem.

Consequences of Theorem 2.4.

  1. 1.

If $g_{1}=g_{2}=g$, then

$\langle\mathcal{W}^{A_{1},A_{2}}_{f_{1},g}(t,u),{\mathcal{W}^{A_{1},A_{2}}_{f_{2},g}(w,u)}\rangle=\|g\|^{2}_{L^{2}(\mathbf{R}^{2})}\langle
f_{1},f_{2}\rangle$ (9)

  2. 2.

If $f_{1}=f_{2}=f$, then

$\langle\mathcal{W}^{A_{1},A_{2}}_{f,g_{1}}(t,u),{\mathcal{W}^{A_{1},A_{2}}_{f,g_{2}}(w,u)}\rangle=\|f\|^{2}_{L^{2}(\mathbf{R}^{2})}\langle
g_{1},g_{2}\rangle.$ (10)

  3. 3.

If $f_{1}=f_{2}=f$ and $g_{1}=g_{2}=g$, then

$\displaystyle\langle\mathcal{W}^{A_{1},A_{2}}_{f,g}(t,u),{\mathcal{W}^{A_{1},A_{2}}_{f,g}(w,u)}\rangle$
$\displaystyle=\int_{\mathbf{R}^{2}}\int_{\mathbf{R}^{2}}|\mathcal{W}^{A_{1},A_{2}}_{f,g}(t,u)|^{2}dudt$
$\displaystyle=\|f\|^{2}_{L^{2}(\mathbf{R}^{2})}\|g\|^{2}_{L^{2}(\mathbf{R}^{2})}$
(11)

###### Theorem 2.5

(Plancherel’s theorem for
WVD-QOLCT).For $f,g\in L^{2}(\mathbf{R}^{2},\mathbf{H})$, we have the equality

$\displaystyle\int_{\mathbf{R}^{2}}\int_{\mathbf{R}^{2}}|\mathcal{W}^{A_{1},A_{2}}_{f,g}(t,u)|^{2}dudt$
$\displaystyle=\|\mathcal{W}^{A_{1},A_{2}}_{f,g}\|^{2}_{L^{2}(\mathbf{R}^{2},\mathbf{H})}$
$\displaystyle=\|f\|^{2}_{L^{2}(\mathbf{R}^{2},\mathbf{H})}\|g|^{2}_{L^{2}(\mathbf{R}^{2},\mathbf{H})}$
(12)

###### Proof

If we look at (11), the proof of the theorem follows.

Now we move forward towards our main section that is convolution and
correlation theorems for winger-ville distribution associated with quaternion
offset linear canonical transform.

##  3  Convolution and Correlation theorem for WVD-QOLCT

The convolution and correlation are fundamental signal processing algorithms
in the theory of linear time-invariant(LTI) systems. In engineering, they have
been widely used for various template matchings. In the following we first
define the convolution and correlation for the QOLCT. They are extensions of
the convolution definition from the OLCT (see 16 ) to the QOLCT domain. We
then establish the new convolution and correlation for the WVD-QOLCT.We also
show that the convolution theorems of the QWVD and WVD-QLCT can be looked as a
special case of our achieved results.

###### Definition 2

For any two quaternion functions $f,g\in L^{2}(\mathbf{R}^{2},\mathbf{H}),$ we
define the convolution operator of the QOLCT as

$(f\star
g)(t)=\int_{\mathbf{R}^{2}}\Psi(z_{1},t_{1})f(z)g(t-z)\Psi(z_{2},t_{2})dz$
(13)

Where $\Psi(z_{1},t_{1})$ and $\Psi(z_{2},t_{2})$ are known as weight
functions.
We assume

$\displaystyle\Psi(z_{1},t_{1})=e^{-i\frac{a_{1}}{b_{1}}2z_{1}(t_{1}-z_{1})}$

and

$\Psi(z_{2},t_{2})=e^{-j\frac{a_{2}}{b_{2}}2z_{2}(t_{2}-z_{2})}$ (14)

As a consequence of the above definition, we get the following important
theorem.

###### Theorem 3.1

(WVD-QOLCT Convolution). For any two quaternion functions $f,g\in
L^{2}{(\mathbf{R}^{2},\mathbf{H})},$ the following result holds

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\star g}(t,u)$
$\displaystyle=\sqrt{2\pi
b_{1}i}e^{\frac{-i}{2b_{1}}[d_{1}(u_{1}^{2}+r_{1}^{2})-2u_{1}(d_{1}r_{1}-b_{1}s_{1})]}$
$\displaystyle\times\left\\{\int_{\mathbf{R}^{2}}e^{-i\frac{a_{1}}{b_{1}}(4w_{1}(t_{1}-w_{1}))}\mathcal{W}^{A_{1},A_{2}}_{f,f}(w,u)\mathcal{W}^{A_{1},A_{2}}_{g,g}(t-w,u)\right.$
$\displaystyle\left.e^{-j\frac{a_{2}}{b_{2}}(4w_{2}(t_{2}-w_{2}))}dw\right\\}$
$\displaystyle\times\sqrt{2\pi
b_{2}j}e^{\frac{-j}{2b_{2}}[d_{2}(u_{2}^{2}+r_{2}^{2})-2u_{2}(d_{2}r_{2}-b_{2}s_{2})]}$
(15)

###### Proof

Applying the definition of the WVD-QOLCT we have

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\star g}(t,u)$ $\displaystyle=$
$\displaystyle\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}\left[(f\star
g)(t+\frac{n}{2})\right]$
$\displaystyle\left[\overline{f}\star\overline{g}(t-\frac{n}{2})\right]K^{j}_{A_{2}}(n_{2},u_{2})dn$
(16)

Now using Definition 2 in (16) we have

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\star g}(t,u)$
$\displaystyle=\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}\left\\{\int_{{\mathbf{R}}^{2}}\Psi_{1}(z_{1},t_{1}+\frac{n_{1}}{2})f(z)\right.$
$\displaystyle\qquad
g(t+\frac{n}{2}-z)\Psi_{2}(z_{2},t_{2}+\frac{n_{2}}{2})dz$
$\displaystyle\qquad\left.\times\int_{{\mathbf{R}}^{2}}\Psi_{1}(\gamma_{1},t_{1}-\frac{n_{1}}{2})\overline{f(\gamma)g(t-\frac{n}{2}-\gamma)}\right.$
$\displaystyle\qquad\left.\Psi_{2}(\gamma_{2},t_{2}-\frac{n_{2}}{2})d\gamma\right\\}{K^{j}_{A_{2}}(n_{2},u_{2})}dn$
$\displaystyle=\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}\left\\{\int_{{\mathbf{R}}^{2}}e^{-i\frac{a_{1}}{b_{1}}2z_{1}((t_{1}+\frac{n_{1}}{2})-z_{1})}f(z)\right.$
$\displaystyle\qquad
g(t+\frac{n}{2}-z)e^{-j\frac{a_{2}}{b_{2}}2z_{2}((t_{2}+\frac{n_{2}}{2})-z_{2})}dz$
$\displaystyle\qquad\times\int_{{\mathbf{R}}^{2}}e^{-i\frac{a_{1}}{b_{1}}2\gamma_{1}((t_{1}-\frac{n_{1}}{2})-\gamma_{1})}\overline{f(\gamma)g(t-\frac{n}{2}-\gamma)}$
$\displaystyle\qquad\left.e^{-j\frac{a_{2}}{b_{2}}2\gamma_{2}((t_{2}-\frac{n_{2}}{2})-\gamma_{2})}d\gamma\right\\}{K^{j}_{A_{2}}(n_{2},u_{2})}dn$
(17)

For simplicity let us denote

$\displaystyle K^{i}_{A_{1}}(t_{1},u_{1})$ $\displaystyle=$ $\displaystyle
K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[a_{1}t_{1}^{2}+2t_{1}(r_{1}-u_{1})-2u_{1}(d_{1}r_{1}-b_{1}s_{1})+d_{1}u_{1}^{2}]},$
$\displaystyle K^{i}_{A_{1}}=\frac{1}{\sqrt{2\pi
b_{1}i}}e^{i\frac{d_{1}}{2b_{1}}r_{1}^{2}}$ (18)

and

$\displaystyle K^{j}_{A_{2}}(t_{2},u_{2})$ $\displaystyle=$ $\displaystyle
K^{j}_{A_{2}}e^{\frac{j}{2b_{2}}[a_{2}t_{2}^{2}+2t_{2}(r_{2}-u_{2})-2u_{2}(d_{2}r_{2}-b_{2}s_{2})+d_{2}u_{2}^{2}]},$
$\displaystyle K^{j}_{A_{2}}=\frac{1}{\sqrt{2\pi
b_{2}j}}e^{j\frac{d_{2}}{2b_{2}}r_{2}^{2}}$ (19)

Now using (18)and (19) in (17),we have

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\star g}(t,u)$
$\displaystyle=\int_{{\mathbf{R}^{6}}}K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[a_{1}n_{1}^{2}+2n_{1}(r_{1}-u_{1})-2u_{1}(d_{1}r_{1}-b_{1}s_{1})+d_{1}u_{1}^{2}]}$
$\displaystyle\;e^{-i\frac{a_{1}}{b_{1}}2z_{1}((t_{1}+\frac{n_{1}}{2})-z_{1})}$
$\displaystyle\;\times
f(z)g(t+\frac{n}{2}-z)e^{-j\frac{a_{2}}{b_{2}}2z_{2}((t_{2}+\frac{n_{2}}{2})-z_{2})}$
$\displaystyle\;e^{-i\frac{a_{1}}{b_{1}}2\gamma_{1}((t_{1}-\frac{n_{1}}{2})-\gamma_{1})}$
$\displaystyle\;\times\overline{f(\gamma)g(t-\frac{n}{2}-\gamma)}e^{-j\frac{a_{2}}{b_{2}}2\gamma_{2}((t_{2}-\frac{n_{2}}{2})-\gamma_{2})}$
$\displaystyle\;\times
K^{j}_{A_{2}}e^{\frac{j}{2b_{2}}[a_{2}n_{2}^{2}+2n_{2}(r_{2}-u_{2})-2u_{2}(d_{2}r_{2}-b_{2}s_{2})+d_{2}u_{2}^{2}]}dzd\gamma
dn$

Setting $z_{i}=w_{i}+\frac{p_{i}}{2},\gamma_{i}=w_{i}-\frac{p_{i}}{2},i=1,2$
we get

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\star g}(t,u)$
$\displaystyle=\int_{{\mathbf{R}^{6}}}K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[a_{1}n_{1}^{2}+2n_{1}(r_{1}-u_{1})-2u_{1}(d_{1}r_{1}-b_{1}s_{1})+d_{1}u_{1}^{2}]}$
$\displaystyle\qquad
e^{-i\frac{a_{1}}{b_{1}}2\left(w_{1}+\frac{p_{1}}{2}\right)\left((t_{1}+\frac{n_{1}}{2})-(w_{1}+\frac{p_{1}}{2})\right)}$
$\displaystyle\qquad\times
f\left(w+\frac{p}{2}\right)g\left(t+\frac{n}{2}-(w+\frac{p}{2})\right)$
$\displaystyle\qquad
e^{-j\frac{a_{2}}{b_{2}}2\left(w_{2}+\frac{p_{2}}{2}\right)\left((t_{2}+\frac{n_{2}}{2})-(w_{2}+\frac{p_{2}}{2})\right)}$
$\displaystyle\qquad\times
e^{-i\frac{a_{1}}{b_{1}}2\left(w_{1}-\frac{p_{1}}{2}\right)\left((t_{1}-\frac{n_{1}}{2})-(w_{1}-\frac{p_{1}}{2})\right)}$
$\displaystyle\qquad\overline{f\left(w-\frac{p}{2}\right)g\left(t-\frac{n}{2}-(w-\frac{p}{2})\right)}$
$\displaystyle\qquad\times
e^{-j\frac{a_{2}}{b_{2}}2\left(w_{2}-\frac{p_{2}}{2}\right)\left((t_{2}-\frac{n_{2}}{2})-(w_{2}-\frac{p_{2}}{2})\right)}$
$\displaystyle\;K^{j}_{A_{2}}e^{\frac{j}{2b_{2}}[a_{2}n_{2}^{2}+2n_{2}(r_{2}-u_{2})-2u_{2}(d_{2}r_{2}-b_{2}s_{2})+d_{2}u_{2}^{2}]}dpdqdw$

and $n_{i}=p_{i}+q_{i},i=1,2$ we obtain

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\star g}(t,u)$
$\displaystyle=\int_{{\mathbf{R}^{6}}}$
$\displaystyle\;K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[a_{1}(p_{1}+q_{1})^{2}+2(p_{1}+q_{1})(r_{1}-u_{1})-2u_{1}(d_{1}r_{1}-b_{1}s_{1})+d_{1}u_{1}^{2}]}$
$\displaystyle\;e^{-i\frac{a_{1}}{b_{1}}\left(4w_{1}(t_{1}-w_{1})\right)}e^{-i\frac{a_{1}}{b_{1}}p_{1}q_{1}}$
$\displaystyle\;\times
f\left(w+\frac{p}{2}\right)\overline{f\left(w-\frac{p}{2}\right)}g\left(t-w+\frac{q}{2}\right)\overline{g\left(t-w-\frac{q}{2}\right)}$
$\displaystyle\;e^{-j\frac{a_{2}}{b_{2}}\left(4w_{2}(t_{2}-w_{2})\right)}e^{-j\frac{a_{2}}{b_{2}}p_{2}q_{2}}$
$\displaystyle\;\times
K^{j}_{A_{2}}e^{\frac{j}{2b_{2}}[a_{2}n_{2}^{2}+2n_{2}(r_{2}-u_{2})-2u_{2}(d_{2}r_{2}-b_{2}s_{2})+d_{2}u_{2}^{2}]}dpdqdw$
$\displaystyle=\int_{{\mathbf{R}^{2}}}\left\\{\left[\int_{{\mathbf{R}^{2}}}\right.\right.$
$\displaystyle\;K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[a_{1}p_{1}^{2}+2p_{1}(r_{1}-u_{1})-2u_{1}(d_{1}r_{1}-b_{1}s_{1})+d_{1}u_{1}^{2}]}$
$\displaystyle\;f\left(w+\frac{p}{2}\right)\overline{f\left(w-\frac{p}{2}\right)}$
$\displaystyle\;\times\left.K^{j}_{A_{2}}e^{\frac{j}{2b_{2}}[a_{2}p_{2}^{2}+2p_{2}(r_{2}-u_{2})-2u_{2}(d_{2}r_{2}-b_{2}s_{2})+d_{2}u_{2}^{2}]}dp\right]$
$\displaystyle\;\times\int_{{\mathbf{R}^{2}}}e^{\frac{i}{2b_{1}}[a_{1}q_{1}^{2}-2q_{1}(r_{1}-u_{1})]}g\left(t-w+\frac{q}{2}\right)$
$\displaystyle\;\times\left.\overline{g\left(t-w-\frac{q}{2}\right)}e^{\frac{j}{2b_{2}}[a_{2}q_{2}^{2}-2q_{2}(r_{2}-u_{2})]}dq\right\\}$
$\displaystyle\;\times
e^{-i\frac{a_{1}}{b_{1}}\left(4w_{1}(t_{1}-w_{1})\right)}e^{-j\frac{a_{2}}{b_{2}}\left(4w_{2}(t_{2}-w_{2})\right)}dw$
(20)

Now multiply (20)both sides by
$K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[d_{1}u_{1}^{2}-2u_{1}(d_{1}r_{1}-b_{1}s_{1})]}$
and
$K^{j}_{A_{2}}e^{\frac{j}{2b_{2}}[d_{2}u_{2}^{2}-2u_{2}(d_{2}r_{2}-b_{2}s_{2})]}$,we
get

$\displaystyle
K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[d_{1}u_{1}^{2}-2u_{1}(d_{1}r_{1}-b_{1}s_{1})]}$
$\displaystyle\;K^{j}_{A_{2}}e^{\frac{j}{2b_{2}}[d_{2}u_{2}^{2}-2u_{2}(d_{2}r_{2}-b_{2}s_{2})]}\mathcal{W}^{A_{1},A_{2}}_{f\star
g}(t,u)$
$\displaystyle=\int_{{\mathbf{R}^{2}}}e^{-i\frac{a_{1}}{b_{1}}\left(4w_{1}(t_{1}-w_{1})\right)}\mathcal{W}^{A_{1},A_{2}}_{f,f}(w,u)$
$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{g,g}(t-w,u)e^{-j\frac{a_{2}}{b_{2}}\left(4w_{2}(t_{2}-w_{2})\right)}dw$
(21)

Now using (18) and (19) in (21) we get,

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\star g}(t,u)$
$\displaystyle=\sqrt{2\pi
b_{1}i}e^{\frac{-i}{2b_{1}}[d_{1}(u_{1}^{2}+r_{1}^{2})-2u_{1}(d_{1}r_{1}-b_{1}s_{1})]}$
$\displaystyle\;\times\left\\{\int_{\mathbf{R}^{2}}e^{-i\frac{a_{1}}{b_{1}}(4w_{1}(t_{1}-w_{1}))}\mathcal{W}^{A_{1},A_{2}}_{f,f}(w,u)\mathcal{W}^{A_{1},A_{2}}_{g,g}(t-w,u)\right.$
$\displaystyle\;\left.e^{-j\frac{a_{2}}{b_{2}}(4w_{2}(t_{2}-w_{2}))}dw\right\\}$
$\displaystyle\;\times\sqrt{2\pi
b_{2}j}e^{\frac{-j}{2b_{2}}[d_{2}(u_{2}^{2}-r_{2}^{2})-2u_{2}(d_{2}r_{2}-b_{2}s_{2})]}$

which completes the proof of theorem.

Consequences of theorem 3.1.

  1. 1.

Changing parameter $A_{i}=\left[\begin{array}[]{cccc}a_{i}&b_{i}&|&r_{i}\\\
c_{i}&d_{i}&|&s_{i}\\\ \end{array}\right]$ ,$i=1,2$ to
$A_{i}=\left[\begin{array}[]{cccc}a_{i}&b_{i}&|&0\\\ c_{i}&d_{i}&|&0\\\
\end{array}\right]$ ,$i=1,2$, then the Theorem 2.4 reduces to convolution
theorem of the WVD-QLCT as follows:

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\star g}(t,u)$
$\displaystyle=\sqrt{2\pi b_{1}i}e^{\frac{-i}{2b_{1}}d_{1}u_{1}^{2}}\sqrt{2\pi
b_{2}j}e^{\frac{-j}{2b_{2}}d_{2}u_{2}^{2}}$
$\displaystyle\times\left\\{\int_{\mathbf{R}^{2}}e^{-i\frac{a_{1}}{b_{1}}(4w_{1}(t_{1}-w_{1}))}\mathcal{W}^{A_{1},A_{2}}_{f,f}(w,u)\mathcal{W}^{A_{1},A_{2}}_{g,g}(t-w,u)\right.$
$\displaystyle\left.e^{-j\frac{a_{2}}{b_{2}}(4w_{2}(t_{2}-w_{2}))}dw\right\\}$
(22)

where $\mathcal{W}^{A_{1},A_{2}}_{f,f}$ and $\mathcal{W}^{A_{1},A_{2}}_{g,g}$
is the WVD in the QLCT domain of a signal $f$ and $g$, respectively.

  2. 2.

Changing parameter $A_{i}=\left[\begin{array}[]{cccc}a_{i}&b_{i}&|&r_{i}\\\
c_{i}&d_{i}&|&s_{i}\\\ \end{array}\right]$ ,$i=1,2$ to
$A_{i}=\left[\begin{array}[]{cccc}0&1&|&0\\\ -1&0&|&0\\\ \end{array}\right]$
,$i=1,2$, then the Theorem 2.4 reduces to convolution theorem of the WVD in
Quaternion Domain as follows:

$\displaystyle\mathcal{W}_{f\star g}(t,u)$ $\displaystyle=$
$\displaystyle\sqrt{2\pi
i}\left\\{\int_{\mathbf{R}^{2}}\mathcal{W}_{f,f}^{i,j}(w,u)\mathcal{W}_{g,g}^{i,j}(t-w,u)dw\right\\}$
$\displaystyle\sqrt{2\pi j}$ (23)

where $\mathcal{W}_{f,f}^{i,j}$ and $\mathcal{W}_{g,g}^{i,j}$ is the WVD in
the Quaternion domain of a signal $f$ and $g$, respectively.

Next, we will derive the correlation theorem in the WVD-QOLCT. Let us define
the correlation for the QOLCT.

###### Definition 3

For any two quaternion functions $f,g\in L^{2}(\mathbf{R}^{2},\mathbf{H}),$ we
define the correlation operator of the QOLCT as

$(f\circ
g)(t)=\int_{\mathbf{R}^{2}}e^{i\frac{a_{1}}{b_{1}}2z_{1}(z_{1}+t_{1})}\overline{f(z)}g(z+t)e^{j\frac{a_{2}}{b_{2}}2z_{2}(z_{2}+t_{2})}dz$
(24)

Now, we reap a consequence of the above definition .

###### Theorem 3.2

(WVD-QOLCT Correlation). For any two quaternion functions $f,g\in
L^{2}{(\mathbf{R}^{2},\mathbf{H})},$ the following result holds

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\circ g}(t,u)$
$\displaystyle=\sqrt{2\pi
b_{1}i}e^{\frac{-i}{2b_{1}}[d_{1}(u_{1}^{2}+r_{1}^{2})+2u_{1}(d_{1}r_{1}-b_{1}s_{1})]}$
$\displaystyle\times\left\\{\int_{\mathbf{R}^{2}}e^{i\frac{a_{1}}{b_{1}}(4w_{1}(t_{1}+w_{1}))}\mathcal{W}^{A_{1},A_{2}}_{f,f}(w,-u)\right.$
$\displaystyle\left.\mathcal{W}^{A_{1},A_{2}}_{g,g}(t+w,u)e^{j\frac{a_{2}}{b_{2}}(4w_{2}(t_{2}+w_{2}))}dw\right\\}$
$\displaystyle\times\sqrt{2\pi
b_{2}j}e^{\frac{-j}{2b_{2}}[d_{2}(u_{2}^{2}+r_{2}^{2})+2u_{2}(d_{2}r_{2}-b_{2}s_{2})]}$
(25)

###### Proof

Applying the definition of the WVD-QOLCT we have

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\circ g}(t,u)$
$\displaystyle=\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}\left[(f\circ
g)(t+\frac{n}{2})\right]$
$\displaystyle\left[\overline{f}\circ\overline{g}(t-\frac{n}{2})\right]K^{j}_{A_{2}}(n_{2},u_{2})dn$
(26)

Now using definition 3 in (26) we have

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\circ g}(t,u)$
$\displaystyle=\int_{{\mathbf{R}}^{2}}{K^{i}_{A_{1}}(n_{1},u_{1})}\left\\{\int_{{\mathbf{R}}^{2}}e^{i\frac{a_{1}}{b_{1}}2z_{1}(z_{1}+(t_{1}+\frac{n_{1}}{2}))}\overline{f(z)}\right.$
$\displaystyle
g(z+t+\frac{n}{2})e^{j\frac{a_{2}}{b_{2}}2z_{2}(z_{2}+(t_{2}+\frac{n_{2}}{2}))}dz$
$\displaystyle\;\times\int_{{\mathbf{R}}^{2}}e^{i\frac{a_{1}}{b_{1}}2\gamma_{1}(\gamma_{1}+(t_{1}-\frac{n_{1}}{2}))}\overline{\overline{f(\gamma)}g(\gamma+(t-\frac{n}{2}))}$
$\displaystyle\left.e^{j\frac{a_{2}}{b_{2}}2\gamma_{2}(\gamma_{2}+(t_{2}-\frac{n_{2}}{2}))}d\gamma\right\\}{K^{j}_{A_{2}}(n_{2},u_{2})}dn$
(27)

Now with the help of (18) and (19),we have from (27)

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\circ g}(t,u)$
$\displaystyle=\int_{{\mathbf{R}^{6}}}K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[a_{1}n_{1}^{2}+2n_{1}(r_{1}-u_{1})-2u_{1}(d_{1}r_{1}-b_{1}s_{1})+d_{1}u_{1}^{2}]}$
$\displaystyle\;e^{i\frac{a_{1}}{b_{1}}2z_{1}(z_{1}+(t_{1}+\frac{n_{1}}{2}))}$
$\displaystyle\;\times\overline{f(z)}g(z+(t+\frac{n}{2}))e^{j\frac{a_{2}}{b_{2}}2z_{2}(z_{2}+(t_{2}+\frac{n_{2}}{2}))}e^{i\frac{a_{1}}{b_{1}}2\gamma_{1}(\gamma_{1}+(t_{1}-\frac{n_{1}}{2}))}$
$\displaystyle\;\times\overline{\overline{f(\gamma)}g(\gamma+(t-\frac{n}{2}))}e^{j\frac{a_{2}}{b_{2}}2\gamma_{2}(\gamma_{2}+(t_{2}-\frac{n_{2}}{2}))}$
$\displaystyle\;\times
K^{j}_{A_{2}}e^{\frac{j}{2b_{2}}[a_{2}n_{2}^{2}+2n_{2}(r_{2}-u_{2})-2u_{2}(d_{2}r_{2}-b_{2}s_{2})+d_{2}u_{2}^{2}]}dzd\gamma
dn$

Setting $z_{i}=w_{i}+\frac{p_{i}}{2},\gamma_{i}=w_{i}-\frac{p_{i}}{2},i=1,2$,
we get

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\circ g}(t,u)$
$\displaystyle=\int_{{\mathbf{R}^{6}}}K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[a_{1}n_{1}^{2}+2n_{1}(r_{1}-u_{1})-2u_{1}(d_{1}r_{1}-b_{1}s_{1})+d_{1}u_{1}^{2}]}$
$\displaystyle
e^{i\frac{a_{1}}{b_{1}}2\left(w_{1}+\frac{p_{1}}{2}\right)\left((t_{1}+\frac{n_{1}}{2})+(w_{1}+\frac{p_{1}}{2})\right)}$
$\displaystyle\;\times\overline{f\left(w+\frac{p}{2}\right)}g\left((t+\frac{n}{2})+(w+\frac{p}{2})\right)$
$\displaystyle
e^{j\frac{a_{2}}{b_{2}}2\left(w_{2}+\frac{p_{2}}{2}\right)\left((t_{2}+\frac{n_{2}}{2})+(w_{2}+\frac{p_{2}}{2})\right)}$
$\displaystyle\;\times
e^{i\frac{a_{1}}{b_{1}}2\left(w_{1}-\frac{p_{1}}{2}\right)\left((t_{1}-\frac{n_{1}}{2})+(w_{1}-\frac{p_{1}}{2})\right)}f\left(w-\frac{p}{2}\right)$
$\displaystyle\overline{g\left((t-\frac{n}{2})+(w-\frac{p}{2})\right)}$
$\displaystyle\times
e^{j\frac{a_{2}}{b_{2}}2\left(w_{2}-\frac{p_{2}}{2}\right)\left((t_{2}-\frac{n_{2}}{2})+(w_{2}-\frac{p_{2}}{2})\right)}K^{j}_{A_{2}}$
$\displaystyle
e^{\frac{j}{2b_{2}}[a_{2}n_{2}^{2}+2n_{2}(r_{2}-u_{2})-2u_{2}(d_{2}r_{2}-b_{2}s_{2})+d_{2}u_{2}^{2}]}dpdqdw$
(28)

Now put $n_{i}=q_{i}-p_{i},i=1,2$ and on following the same procedure as
followed in previous Theorem 3.1, we have from (28)

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\circ g}(t,u)$
$\displaystyle=\int_{{\mathbf{R}^{2}}}\left[\int_{{\mathbf{R}^{2}}}e^{\frac{i}{2b_{1}}[a_{1}p_{1}^{2}-2p_{1}(r_{1}-u_{1})]}\overline{f\left(w+\frac{p}{2}\right)}f\left(w-\frac{p}{2}\right)\right.$
$\displaystyle\left.e^{\frac{j}{2b_{2}}[a_{2}p_{2}^{2}-2p_{2}(r_{2}-u_{2})]}dp\right]$
$\displaystyle\times\left[\int_{{\mathbf{R}^{2}}}K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[a_{1}q_{1}^{2}+2q_{1}(r_{1}-u_{1})-2u_{1}(d_{1}r_{1}-b_{1}s_{1})+d_{1}u_{1}^{2}]}\right.$
$\displaystyle\times
g\left(t+w+\frac{q}{2}\right)\overline{g\left(t+w-\frac{q}{2}\right)}$
$\displaystyle\times\left.K^{j}_{A_{2}}e^{\frac{j}{2b_{2}}[a_{2}q_{2}^{2}+2q_{2}(r_{2}-u_{2})-2u_{2}(d_{2}r_{2}-b_{2}s_{2})+d_{2}u_{2}^{2}]}dq\right]$
$\displaystyle\times
e^{i\frac{a_{1}}{b_{1}}\left(4w_{1}(t_{1}+w_{1})\right)}e^{j\frac{a_{2}}{b_{2}}\left(4w_{2}(t_{2}+w_{2})\right)}dw$
(29)

On multiplying (29) both sides by
$K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[d_{1}u_{1}^{2}-2u_{1}(d_{1}r_{1}-b_{1}s_{1})+4p_{1}(r_{1}-u_{1})]}$
and
$K^{j}_{A_{2}}e^{\frac{j}{2b_{2}}[d_{2}u_{2}^{2}-2u_{2}(d_{2}r_{2}-b_{2}s_{2})+4p_{2}(r_{2}-u_{2})]}$,we
get

$\displaystyle
K^{i}_{A_{1}}e^{\frac{i}{2b_{1}}[d_{1}u_{1}^{2}-2u_{1}(d_{1}r_{1}-b_{1}s_{1})+4p_{1}(r_{1}-u_{1})]}$
$\displaystyle
K^{j}_{A_{2}}e^{\frac{j}{2b_{2}}[d_{2}u_{2}^{2}-2u_{2}(d_{2}r_{2}-b_{2}s_{2})+4p_{2}(r_{2}-u_{2})]}\mathcal{W}^{A_{1},A_{2}}_{f\circ
g}(t,u)$
$\displaystyle=\int_{{\mathbf{R}^{2}}}e^{-i\frac{a_{1}}{b_{1}}\left(4w_{1}(t_{1}-w_{1})\right)}\mathcal{W}^{A_{1},A_{2}}_{f,f}(w,u)$
$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{g,g}(t-w,u)e^{-j\frac{a_{2}}{b_{2}}\left(4w_{2}(t_{2}-w_{2})\right)}dw$
(30)

Now using (18) and (19) in (30) we obtain,

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\star g}(t,u)$
$\displaystyle=\sqrt{2\pi
b_{1}i}e^{\frac{-i}{2b_{1}}[d_{1}(u_{1}^{2}+r_{1}^{2})+2u_{1}(d_{1}r_{1}-b_{1}s_{1})]}$
$\displaystyle\times\left\\{\int_{\mathbf{R}^{2}}e^{i\frac{a_{1}}{b_{1}}(4w_{1}(t_{1}+w_{1}))}\mathcal{W}^{A_{1},A_{2}}_{f,f}(w,-u)\mathcal{W}^{A_{1},A_{2}}_{g,g}(t+w,u)\right.$
$\displaystyle\times\left.e^{j\frac{a_{2}}{b_{2}}(4w_{2}(t_{2}+w_{2}))}dw\right\\}\sqrt{2\pi
b_{2}j}e^{\frac{-j}{2b_{2}}[d_{2}(u_{2}^{2}+r_{2}^{2})+2u_{2}(d_{2}r_{2}-b_{2}s_{2})]}$

which completes the proof of theorem.

Consequences of Theorem 3.2.

  1. 1.

Changing parameter $A_{i}=\left[\begin{array}[]{cccc}a_{i}&b_{i}&|&r_{i}\\\
c_{i}&d_{i}&|&s_{i}\\\ \end{array}\right]$ ,$i=1,2$ to
$A_{i}=\left[\begin{array}[]{cccc}a_{i}&b_{i}&|&0\\\ c_{i}&d_{i}&|&0\\\
\end{array}\right]$ ,$i=1,2$, then the Theorem 3.2 reduces to correlation
theorem of the WVD-QLCT as follows:

$\displaystyle\mathcal{W}^{A_{1},A_{2}}_{f\circ g}(t,u)$
$\displaystyle=\sqrt{2\pi b_{1}i}e^{\frac{-i}{2b_{1}}d_{1}u_{1}^{2}}\sqrt{2\pi
b_{2}j}e^{\frac{-j}{2b_{2}}d_{2}u_{2}^{2}}$
$\displaystyle\times\left\\{\int_{\mathbf{R}^{2}}e^{i\frac{a_{1}}{b_{1}}(4w_{1}(t_{1}+w_{1}))}\mathcal{W}^{A_{1},A_{2}}_{f,f}(w,-u)\right.$
$\displaystyle\left.\mathcal{W}^{A_{1},A_{2}}_{g,g}(t+w,u)e^{j\frac{a_{2}}{b_{2}}(4w_{2}(t_{2}+w_{2}))}dw\right\\}$

where $\mathcal{W}^{A_{1},A_{2}}_{f,f}$ and $\mathcal{W}^{A_{1},A_{2}}_{g,g}$
is the WVD in the QLCT domain of a signal $f$ and $g$, respectively.

  2. 2.

Changing parameter $A_{i}=\left[\begin{array}[]{cccc}a_{i}&b_{i}&|&r_{i}\\\
c_{i}&d_{i}&|&s_{i}\\\ \end{array}\right]$ ,$i=1,2$ to
$A_{i}=\left[\begin{array}[]{cccc}0&1&|&0\\\ -1&0&|&0\\\ \end{array}\right]$
,$i=1,2$, then the Theorem 4.4 reduces to correlation theorem of the WVD in
Quaternion Domain as follows:

$\displaystyle\mathcal{W}_{f\circ g}(t,u)$ $\displaystyle=$
$\displaystyle\sqrt{2\pi
i}\left\\{\int_{\mathbf{R}^{2}}\mathcal{W}_{f,f}^{i,j}(w,-u)\mathcal{W}_{g,g}^{i,j}(t+w,u)dw\right\\}\sqrt{2\pi
j}$

where $\mathcal{W}_{f,f}^{i,j}$ and $\mathcal{W}_{g,g}^{i,j}$ is the WVD in
the Quaternion domain of a signal $f$ and $g$, respectively.

###### Acknowledgements.

This work is supported by the UGC-BSR Research Start Up Grant(No.
F.30-498/2019(BSR)) provided by UGC, Govt. of India.

## Conflict of interest

The authors declare that they have no conflict of interest.

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