# Integral representation of superoscillations
via complex Borel measures and their convergence

Jussi Behrndt  (JB) Institute of Applied Mathematics, Graz University of
Technology, Steyrergasse 30, 8010 Graz, Austria<EMAIL_ADDRESS>,  Fabrizio
Colombo  (FC) Politecnico di Milano, Dipartimento di Matematica, Via E.
Bonardi, 9, 20133 Milano, Italy<EMAIL_ADDRESS>,  Peter Schlosser
(PS) Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9,
20133 Milano, Italy<EMAIL_ADDRESS>and  Daniele C. Struppa  (DCS)
The Donald Bren Presidential Chair in Mathematics, Chapman University, Orange,
USA<EMAIL_ADDRESS>
###### Abstract.

In the last decade there has been a growing interest in superoscillations in
various fields of mathematics, physics and engineering. However, while in
applications as optics the local oscillatory behaviour is the important
property, some convergence to a plane wave is the standard characterizing
feature of a superoscillating function in mathematics and quantum mechanics.
Also there exists a certain discrepancy between the representation of
superoscillations either as generalized Fourier series, as certain integrals
or via special functions. The aim of this work is to close these gaps and give
a general definition of superoscillations, covering the well-known examples in
the existing literature. Superoscillations will be defined as sequences of
holomorphic functions, which admit integral representations with respect to
complex Borel measures and converge to a plane wave in the space
$\mathcal{A}_{1}(\mathbb{C})$ of exponentially bounded entire functions.

AMS Classification 35A20, 35A08.

Keywords: Superoscillations, Borel measures, infinite order differential
operators.

##  1\. Introduction

The theory of superoscillatory functions has its origin in various areas of
physics and engineering. For example in quantum mechanics these functions
appear in the context of Aharonov’s weak values [2], in antenna theory they
were first used in [33] and for optics and other applications we refer to the
Roadmap on Superoscillations by M.V. Berry et al. [19].

The prototypical superoscillatory function that appeared in quantum mechanics
can be found in the still unpublished preprint [13] from 1991, where Y.
Aharonov and collaborators made a thought experiment considering a box
containing only red light, but emitting a gamma ray. With the notion of weak
values the authors also found a way How the result of a measurement of a
component of the spin of a spin-$\frac{1}{2}$ particle can turn out to be 100,
see [2]. The type of functions considered in those papers are of the form

$F_{n}(x)=\sum\limits_{j=0}^{n}C_{j}(n)e^{ik_{j}(n)x},\quad
x\in\mathbb{R},\,n\in\mathbb{N},$ (1.1)

where for some fixed $a>1$ the coefficients are chosen as

$C_{j}(n)=\Big{(}\begin{matrix}n\\\
j\end{matrix}\Big{)}\Big{(}\frac{1+a}{2}\Big{)}^{n-j}\Big{(}\frac{1-a}{2}\Big{)}^{j}\quad\text{and}\quad
k_{j}(n)=1-\frac{2j}{n}.$ (1.2)

Note that every $F_{n}$ is a linear combination of plane waves with
frequencies $k_{j}(n)\in[-1,1]$. The superoscillatory behaviour now comes from
the fact that

$\lim\limits_{n\rightarrow\infty}F_{n}(x)=e^{iax},\quad x\in\mathbb{R},$ (1.3)

converges to a plane wave with frequency $a>1$. However, superoscillations of
the form (1.1) do not only appear with the frequencies $k_{j}(n)$ in (1.2),
but it was shown in [6] that for any choice of frequencies $k_{j}(n)\in[-1,1]$
and any $a>1$ one can construct coefficients $C^{\prime}_{j}(n)\in\mathbb{C}$,
such that

$\sum\limits_{j=0}^{n}C_{j}^{\prime}(n)e^{ik_{j}(n)x}\sim e^{iax},\quad
x\in\mathbb{R},$ (1.4)

where $\sim$ is understood in the sense that the Taylor coefficients of both
sides coincide up to order $n$.

The mathematical theory of superoscillations has attracted a lot of interest
in the recent past. A first introduction to the mathematics of
superoscillations in one variable and some of its applications to Schrödinger
evolution of superoscillatory initial data can be found in [9]. Nowadays, the
literature on superoscillations is growing quite fast and, without claiming
completeness, we mention that some of the most recent results on the time
evolution of superoscillations are contained in the papers [3, 4, 5, 16, 30,
31] and in the references therein. The case of superoscillating functions in
several variables and other interesting mathematical aspects of
superoscillations are contained in the papers [8, 11, 12, 22, 23].

A certain type of superoscillations, different from the ones in (1.1), were
introduced by M.V. Berry in his work Faster than Fourier [17], where families
of superoscillating functions are constructed that have the form

$F_{\delta}(x)=\frac{1}{\delta\sqrt{2\pi}}\int_{\mathbb{R}}e^{ik(u)x}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du,\quad
x\in\mathbb{R},\,\delta>0,$ (1.5)

with some fixed coefficient $a>0$ and a frequency function $k$ with values
$k(u)\in[-1,1]$ for every $u\in\mathbb{R}$. Instead of a convergence result of
the form (1.3) it is shown in Berry’s work that the local wavenumber exceeds
the range $[-1,1]$, while the intrinsic frequencies $k$ take only values in
$[-1,1]$. In the same way also the superoscillatory behaviour of the function

$F_{\delta}(x)=\frac{2}{\delta}e^{-\frac{1}{\delta}}\operatorname{sinc}\bigg{(}\sqrt{x^{2}-\frac{2iax}{\delta}-\frac{1}{\delta^{2}}}\bigg{)}\quad
x\in\mathbb{R},\,\delta>0,$ (1.6)

was investigated in [18].

A method of constructing superoscillating functions was introduced in [7] and
revisited in [14]. The idea is to consider for some entire function
$H(z)=\sum_{n=0}^{\infty}h_{n}z^{n}$ the associated infinite order
differential operator $H(-i\frac{\partial}{\partial
x})\coloneqq\sum_{n=0}^{\infty}h_{n}(-i\frac{\partial}{\partial x})^{n}$ and
the corresponding generalized Schrödinger equation

$i\frac{\partial}{\partial t}\Psi(t,x)=-H\Big{(}-i\frac{\partial}{\partial
x}\Big{)}\Psi(t,x),\quad t,x\in\mathbb{R}.$ (1.7)

Using superoscillating initial conditions it turns out that the solution
$\Psi(t,\,\cdot\,)$ at any time $t\in\mathbb{R}$ is still superoscillating.
Moreover, under some additional assumptions on the range of the function $H$
the solution $\Psi(\,\cdot\,,x)$ is also superoscillating in the time variable
for every fixed $x\in\mathbb{R}$.

Furthermore, a different perspective on superoscillatory behaviour was taken
by P. Ferreira, A. Kempf and D. Lee in a series of papers [24, 25, 26, 27, 28,
29]. There functions of the form

$F(x)=\sum\limits_{l=0}^{m}c_{l}\operatorname{sinc}(x-x_{l}),\quad
x\in\mathbb{R},$ (1.8)

were considered, where the coefficients $(c_{l})_{l}$ are chosen such that
$F(x_{l})=a_{l}$, $l\in\\{0,\dots,m\\}$, admits the prescribed values
$(a_{l})_{l}$ at the prescibed points $(x_{l})_{l}$. Choosing now for example
$x_{l}=\delta l$ for some arbitrary small $\delta>0$ and $a_{l}=(-1)^{l}$ with
alternating sign, the function $F$ admits an arbitrary large number of
oscillations in an arbitrary small interval, while its Fourier transform is
always supported in the bounded interval $[-1,1]$.

The above examples show that there exist many variants of superoscillations,
defined in various ways and having different types of oscillatory behaviour.
One of the main purposes of this paper is to propose the general Definition
1.2 of superoscillations below and to show that all the above examples are
contained as special cases in this concept. The crucial points in our analysis
are the integral representation (1.11) via complex Borel measures and the
space $\mathcal{A}_{1}(\mathbb{C})$ of exponentially bounded entire functions,
where the convergence of superoscillating functions is considered with respect
to the natural topology.

######  Definition 1.1.

The space of entire functions with exponential growth is defined as

$\mathcal{A}_{1}(\mathbb{C})\coloneqq\Set{F:\mathbb{C}\rightarrow\mathbb{C}\text{
entire}}{\exists A,B\geq 0\text{ such that }|F(z)|\leq Ae^{B|z|}\text{ for all
}z\in\mathbb{C}}.$ (1.9)

For any $F_{0},(F_{n})_{n}\in\mathcal{A}_{1}(\mathbb{C})$ we say that
$F_{n}\rightarrow F_{0}$ converges in $\mathcal{A}_{1}(\mathbb{C})$ if and
only if there exists some $B\geq 0$, such that

$\lim\limits_{n\rightarrow\infty}\sup\limits_{z\in\mathbb{C}}|F_{n}(z)-F_{0}(z)|e^{-B|z|}=0.$
(1.10)

The idea is now to consider superoscillations as certain superpositions of
plane waves with frequencies in a bounded range $[-k_{0},k_{0}]$, but
converging to a plane wave with frequency
$a\in\mathbb{R}\setminus[-k_{0},k_{0}]$ exceeding this range.

######  Definition 1.2.

A sequence of functions of the form

$F_{n}(z)=\int_{-k_{0}}^{k_{0}}e^{ikz}d\mu_{n}(k),\quad z\in\mathbb{C},$
(1.11)

with a common maximal frequency $k_{0}>0$ and complex Borel measures $\mu_{n}$
on $[-k_{0},k_{0}]$, is called superoscillating, if there exists some
$a\in\mathbb{R}\setminus[-k_{0},k_{0}]$, such that

$\lim\limits_{n\rightarrow\infty}F_{n}(z)=e^{iaz}\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}).$ (1.12)

Note, that any function of the form (1.11) is an element in
$\mathcal{A}_{1}(\mathbb{C})$. In fact, since a complex measure has finite
total variation, the exponential boundedness follows immediately from the
estimate

$|F_{n}(z)|\leq\int_{-k_{0}}^{k_{0}}e^{|k||z|}d|\mu_{n}|(z)\leq|\mu_{n}|([-k_{0},k_{0}])e^{k_{0}|z|},\quad
z\in\mathbb{C}.$

Moreover, the holomorphicity follows from a version of the dominated
convergence theorem which allows to carry the derivatives inside the integral.

Also note that the functions (1.1) are superoscillation in the sense of
Definition 1.2 since the representation (1.11) holds with the complex Borel
measures

$\mu_{n}(B)\coloneqq\sum\limits_{j=0,k_{j}(n)\in B}^{n}C_{j}(n),\quad\text{for
every Borel set }B\subseteq[-1,1],$ (1.13)

and the convergence (1.12) was shown in [21, Theorem 2.1].

In the main part of this paper we collect and generalize the different
superoscillating functions that appear in the mathematical and physical
literature, and prove that they fit into the context of Definition 1.2. In
particular, we verify the convergence in the space
$\mathcal{A}_{1}(\mathbb{C})$ in all cases, which is important from an
analytic point of view and puts the phenomenon of superoscillations in an
appropriate mathematical perspective. In Section 2 functions of the form (1.5)
and certain extensions of these classes are discussed. The superoscillating
$\operatorname{sinc}$-function (1.6) is treated in Section 3. Section 4 shows
that the approximation (1.4) really leads to superoscillations in the sense of
Definition 1.2 for a large class of given frequencies $k_{j}(n)$, see
Corollary 4.2. Moreover, in Theorem 4.1 this concept is generalized to any low
frequency function of the form (4.3) instead of the exponentials
$e^{ik_{j}(n)z}$. It then turns out that a version of the method (1.8) can be
considered as a special case of Corollary 4.4, treated in Example 4.5. The
final Section 5 then considers the method of generating superoscillating
functions described in (1.7).

Acknowledgements. Jussi Behrndt gratefully acknowledges financial support by
the Austrian Science Fund (FWF): P 33568-N. Peter Schlosser’s research was
funded by the Austrian Science Fund (FWF) under Grant No. J 4685-N and by the
European Union – NextGenerationEU. This publication is also based upon work
from COST Action CA 18232 MAT-DYN-NET, supported by COST (European Cooperation
in Science and Technology), www.cost.eu.

##  2\. A construction of superoscillations due to M.V. Berry

The considerations in this section are inspired by M.V. Berry’s paper [17],
where superoscillatory functions of the form

$F_{\delta}(x)=\frac{1}{\delta\sqrt{2\pi}}\int_{\mathbb{R}}e^{ik(u)x}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du,\quad
x\in\mathbb{R},$ (2.1)

were introduced. The idea behind this construction is that the complex
Gaussian

$\frac{1}{\delta\sqrt{2\pi}}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}\rightarrow\delta(u-ia),\quad\text{as
}\delta\rightarrow 0^{+},$

approximates the complex delta function and consequently

$F_{\delta}(x)\rightarrow e^{ik(ia)x},\quad\text{as }\delta\rightarrow 0^{+},$

converges to a plane wave. If one chooses a frequency function with values
$k(u)\in[-1,1]$ for every $u\in\mathbb{R}$ and
$k(ia)\in\mathbb{R}\setminus[-1,1]$ for some $a>0$, this indicates a
superoscillatory behaviour of the functions $F_{\delta}$.

We will now revisit this idea and improve it in three ways. Firstly, an
additional function $g$ is included in the integral (2.1). This function does
not affect the superoscillatory property of the $F_{\delta}$’s, but allows to
modify their shape. Secondly, precise assumptions on the involved functions
$g$ and $k$ are given. Thirdly and most importantly, while in [17] mainly the
complex saddle point approximation is used to derive quantities like the local
wave number, the convergence in the space $\mathcal{A}_{1}(\mathbb{C})$ is
proven here. In particular, Corollary 2.2 shows that the functions
$F_{\delta}$ are indeed superoscillating in the sense of Definition 1.2.

######  Theorem 2.1.

For $b\geq a>0$ let $\Delta\subseteq\mathbb{C}$ be the open triangle with
corners $ia$ and $\pm b$. Consider measurable functions
$k,g:\mathbb{R}\cup\overline{\Delta}\rightarrow\mathbb{C}$ which for some
$B\geq 0$ satisfy the bounds

$\sup\limits_{u\in\mathbb{R}}|k(u)|<\infty\quad\text{and}\quad\sup\limits_{u\in\mathbb{R}}|g(u)|e^{-B|u|}<\infty.$
(2.2)

Moreover, assume that $k$ and $g$ are continuous on $\overline{\Delta}$,
holomorphic on $\Delta$, and in the case $a=b$ their derivatives are bounded
on $\Delta$. Then, for every $\delta>0$, the functions

$F_{\delta}(z)\coloneqq\frac{1}{\delta\sqrt{2\pi}}\int_{\mathbb{R}}g(u)e^{ik(u)z}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du,\quad
z\in\mathbb{C},$ (2.3)

belong to the space $\mathcal{A}_{1}(\mathbb{C})$ and converge as

$\lim\limits_{\delta\rightarrow
0^{+}}F_{\delta}(z)=g(ia)e^{ik(ia)z}\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}).$ (2.4)

###### Proof.

We shall make use of the constants

$C\coloneqq\sup\limits_{u\in\mathbb{R}}|k(u)|\quad\text{and}\quad
A\coloneqq\sup\limits_{u\in\mathbb{R}}|g(u)|e^{-B|u|},$

which are finite due to assumption (2.2). A straightforward estimate shows

$|F_{\delta}(z)|\leq\frac{1}{\delta\sqrt{2\pi}}\int_{\mathbb{R}}|g(u)|e^{|k(u)||z|}e^{\frac{-u^{2}+a^{2}}{2\delta^{2}}}du\leq\frac{A}{\delta\sqrt{2\pi}}e^{\frac{a^{2}}{2\delta^{2}}}\bigg{(}\int_{\mathbb{R}}e^{B|u|}e^{-\frac{u^{2}}{2\delta^{2}}}du\bigg{)}e^{C|z|},$
(2.5)

and hence $F_{\delta}$ is well defined for every $\delta>0$. Furthermore, a
version of the dominated convergence theorem shows that $F_{\delta}$ is
holomorphic and together with the above estimate it follows that $F_{\delta}$,
$\delta>0$, belongs to the space $\mathcal{A}_{1}(\mathbb{C})$.

In the following we will verify that $F_{\delta}$ converges as in (2.4).
First, we change the part $-b\rightarrow b$ of the integration path in (2.3)
to $-b\rightarrow ia\rightarrow b$, that is, we use

$\int_{-b}^{b}g(u)e^{ik(u)z}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du=\int_{-b}^{ia}g(u)e^{ik(u)z}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du+\int_{ia}^{b}g(u)e^{ik(u)z}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du.$

In fact, since $g$ and $k$ are holomorphic on $\Delta$ and continuous on
$\overline{\Delta}$ the above equality follows from the Cauchy theorem applied
to the boundary of the scaled triangle $\Delta_{\varepsilon}$ and then taking
the limit $\varepsilon\rightarrow 0^{+}$.

$\operatorname{Re}$$\operatorname{Im}$-$b$$b$$ia$$\varepsilon$$\varepsilon$$\varepsilon$$\Delta_{\varepsilon}$

Hence we can split up the function $F_{\delta}$ into the four parts

$F_{\delta}(z)=F_{\delta}^{(1)}(z)+F_{\delta}^{(2)}(z)+F_{\delta}^{(3)}(z)+F_{\delta}^{(4)}(z),\quad
z\in\mathbb{C},$ (2.6)

where

$\begin{split}F_{\delta}^{(1)}(z)&=\frac{1}{\delta\sqrt{2\pi}}\int_{-\infty}^{-b}g(u)e^{ik(u)z}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du,\\\
F_{\delta}^{(2)}(z)&=\frac{1}{\delta\sqrt{2\pi}}\int_{-b}^{ia}g(u)e^{ik(u)z}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du,\\\
F_{\delta}^{(3)}(z)&=\frac{1}{\delta\sqrt{2\pi}}\int_{ia}^{b}g(u)e^{ik(u)z}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du,\\\
F_{\delta}^{(4)}(z)&=\frac{1}{\delta\sqrt{2\pi}}\int_{b}^{\infty}g(u)e^{ik(u)z}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du.\end{split}$

For these four functions we will now investigate the limit $\delta\rightarrow
0^{+}$ separately. Starting with $F_{\delta}^{(4)}$ we estimate in the same
way as in (2.5)

$|F_{\delta}^{(4)}(z)|\leq\frac{A}{\delta\sqrt{2\pi}}e^{C|z|}e^{\frac{a^{2}}{2\delta^{2}}}\int_{b}^{\infty}e^{Bu}e^{-\frac{u^{2}}{2\delta^{2}}}du\leq\frac{A}{\sqrt{2\pi}}e^{C|z|}e^{bB}\int_{0}^{\infty}e^{-\frac{v^{2}}{2}}e^{v(B\delta-\frac{b}{\delta})}dv,$

where in the second inequality we used $a\leq b$ and substituted
$v=\frac{u-b}{\delta}$. Shifting the exponential $e^{C|z|}$ to the left side
of this inequality leads to a $z$-independent right hand side, which converges
as

$\sup\limits_{z\in\mathbb{C}}|F_{\delta}^{(4)}(z)|e^{-C|z|}\leq\frac{A}{\sqrt{2\pi}}e^{bB}\int_{0}^{\infty}e^{-\frac{v^{2}}{2}}e^{v(B\delta-\frac{b}{\delta})}dv\rightarrow
0,\quad\text{as }\delta\rightarrow 0^{+}.$

According to (1.10) this is exactly the convergence

$\lim\limits_{\delta\rightarrow 0^{+}}F_{\delta}^{(4)}(z)=0\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}).$

In the same way we also prove

$\lim\limits_{\delta\rightarrow 0^{+}}F_{\delta}^{(1)}(z)=0\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}).$

For the function $F_{\delta}^{(3)}(z)$ we first use the identity

$\int_{ia}^{b}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du=\frac{\delta\sqrt{\pi}}{\sqrt{2}}\operatorname{erf}\Big{(}\frac{b-ia}{\delta\sqrt{2}}\Big{)},$

which is an immediate consequence of the definition of the error function, to
rewrite the difference

$F_{\delta}^{(3)}(z)-\frac{g(ia)}{2}e^{ik(ia)z}=\frac{1}{\delta\sqrt{2\pi}}\int_{ia}^{b}\Big{(}g(u)e^{ik(u)z}-\frac{g(ia)}{\operatorname{erf}\big{(}\frac{b-ia}{\delta\sqrt{2}}\big{)}}e^{ik(ia)z}\Big{)}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du.$
(2.7)

In the case $b>a$ we use $u_{\delta}(t)\coloneqq ia+\delta t(b-ia)$,
$t\in[0,\frac{1}{\delta}]$, to parametrize this integral as

$F_{\delta}^{(3)}(z)-\frac{g(ia)}{2}e^{ik(ia)z}=\frac{b-ia}{\sqrt{2\pi}}\int_{0}^{\frac{1}{\delta}}\Big{(}g(u_{\delta}(t))e^{ik(u_{\delta}(t))z}-\frac{g(ia)}{\operatorname{erf}\big{(}\frac{b-ia}{\delta\sqrt{2}}\big{)}}e^{ik(ia)z}\Big{)}e^{-\frac{t^{2}(b-ia)^{2}}{2}}dt,$

and estimate the difference by

$\displaystyle\Big{|}F_{\delta}^{(3)}(z)-\frac{g(ia)}{2}e^{ik(ia)z}\Big{|}\leq\frac{\sqrt{b^{2}+a^{2}}}{\sqrt{2\pi}}\int_{0}^{\frac{1}{\delta}}\bigg{(}$
$\displaystyle|g(u_{\delta}(t))|\,\big{|}e^{ik(u_{\delta}(t))z}-e^{ik(ia)z}\big{|}$
$\displaystyle+\bigg{|}g(u_{\delta}(t))-\frac{g(ia)}{\operatorname{erf}(\frac{b-ia}{\delta\sqrt{2}})}\bigg{|}e^{|k(ia)|\,|z|}\bigg{)}e^{-\frac{t^{2}(b^{2}-a^{2})}{2}}dt.$

Since $g,k$ are continuous on $\overline{\Delta}$, we denote their respective
suprema by $\|\cdot\|_{\Delta}$. Using the estimate (A.1) we obtain

$\displaystyle\Big{|}F_{\delta}^{(3)}(z)-\frac{g(ia)}{2}e^{ik(ia)z}\Big{|}\leq\frac{\sqrt{b^{2}+a^{2}}}{\sqrt{2\pi}}e^{\|k\|_{\Delta}|z|}\int_{0}^{\frac{1}{\delta}}\bigg{(}$
$\displaystyle|k(u_{\delta}(t))-k(ia)|\,\|g\|_{\Delta}|z|$
$\displaystyle+\bigg{|}g(u_{\delta}(t))-\frac{g(ia)}{\operatorname{erf}(\frac{b-ia}{\delta\sqrt{2}})}\bigg{|}\bigg{)}e^{-\frac{t^{2}(b^{2}-a^{2})}{2}}dt.$

Using also the inequality $|z|\leq e^{|z|}$, which is an immediate consequence
of the power series representation of the exponential, we find the estimate

$\displaystyle\Big{|}F_{\delta}^{(3)}(z)-\frac{g(ia)}{2}e^{ik(ia)z}\Big{|}e^{-(\|k\|_{\Delta}+1)|z|}\leq\frac{\sqrt{b^{2}+a^{2}}}{\sqrt{2\pi}}\int_{0}^{\frac{1}{\delta}}$
$\displaystyle\bigg{(}|k(u_{\delta}(t))-k(ia)|\,\|g\|_{\Delta}$
$\displaystyle+\bigg{|}g(u_{\delta}(t))-\frac{g(ia)}{\operatorname{erf}(\frac{b-ia}{\delta\sqrt{2}})}\bigg{|}\bigg{)}e^{-\frac{t^{2}(b^{2}-a^{2})}{2}}dt.$

In this form we note that the right hand side is independent of $z$ and
vanishes as $\delta\rightarrow 0^{+}$ due to the dominated convergence
theorem, the continuity of $g$ and $k$ and the fact that
$\lim_{\delta\rightarrow
0^{+}}\operatorname{erf}(\frac{b-ia}{\delta\sqrt{2}})=1$ by the choice $b>a$.
This shows the convergence

$\lim\limits_{\delta\rightarrow
0^{+}}F_{\delta}^{(3)}(z)=\frac{g(ia)}{2}e^{ik(ia)z}\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}).$

In the case $a=b$ we go back to formula (2.7) and note that all functions in
the integrand are continuous. Hence we can shift the integration path into the
open triangle $\Delta$ where $g$ and $k$ are holomorphic, i.e. we write the
integral as the limit

$F_{\delta}^{(3)}(z)-\frac{g(ia)}{2}e^{ik(ia)z}=\frac{1}{\delta\sqrt{2\pi}}\lim\limits_{\varepsilon\rightarrow
0^{+}}\int_{i(a-\varepsilon)}^{a-\varepsilon}\Big{(}g(u)e^{ik(u)z}-\frac{g(ia)}{\operatorname{erf}\big{(}\frac{a(1-i)}{\delta\sqrt{2}}\big{)}}e^{ik(ia)z}\Big{)}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du.$

Using the complementary error function $\operatorname{erfc}(z)\coloneqq
1-\operatorname{erf}(z)$ and its derivative

$\frac{d}{du}\operatorname{erfc}\Big{(}\frac{u-ia}{\delta\sqrt{2}}\Big{)}=-\frac{\sqrt{2}}{\delta\sqrt{\pi}}e^{-\frac{(u-ia)^{2}}{2\delta^{2}}},$

integration by parts leads to

$\displaystyle F_{\delta}^{(3)}(z)-\frac{g(ia)}{2}e^{ik(ia)z}$
$\displaystyle\quad=-\frac{1}{2}\lim\limits_{\varepsilon\rightarrow
0^{+}}\int_{i(a-\varepsilon)}^{a-\varepsilon}\Big{(}g(u)e^{ik(u)z}-\frac{g(ia)}{\operatorname{erf}\big{(}\frac{a(1-i)}{\delta\sqrt{2}}\big{)}}e^{ik(ia)z}\Big{)}\frac{d}{du}\operatorname{erfc}\Big{(}\frac{u-ia}{\delta\sqrt{2}}\Big{)}du$
$\displaystyle\quad=-\frac{g(a)}{2}e^{ik(a)z}\operatorname{erfc}\Big{(}\frac{a(1-i)}{\delta\sqrt{2}}\Big{)}+\frac{1}{2}\lim\limits_{\varepsilon\rightarrow
0^{+}}\int_{i(a-\varepsilon)}^{a-\varepsilon}\frac{d}{du}\Big{(}g(u)e^{ik(u)z}\Big{)}\operatorname{erfc}\Big{(}\frac{u-ia}{\delta\sqrt{2}}\Big{)}du$
$\displaystyle\quad=-\frac{g(a)}{2}e^{ik(a)z}\operatorname{erfc}\Big{(}\frac{a(1-i)}{\delta\sqrt{2}}\Big{)}+\frac{1}{2}\lim\limits_{\varepsilon\rightarrow
0^{+}}\int_{0}^{a-\varepsilon}\frac{d}{ds}\Big{(}g(u_{\varepsilon}(s))e^{ik(u_{\varepsilon}(s))z}\Big{)}\operatorname{erfc}\Big{(}\frac{u_{\varepsilon}(s)-ia}{\delta\sqrt{2}}\Big{)}ds,$

where in the last line we parametrized the complex path integral by
$u_{\varepsilon}(s)\coloneqq i(a-\varepsilon)+s(1-i)$,
$s\in[0,a-\varepsilon]$. Using this, we can now estimate the difference by

$\displaystyle\Big{|}F_{\delta}^{(3)}(z)-\frac{g(ia)}{2}e^{ik(ia)z}\Big{|}$
$\displaystyle\leq
e^{\|k\|_{\Delta}|z|}\bigg{(}\frac{\|g\|_{\Delta}}{2}\Big{|}\operatorname{erfc}\Big{(}\frac{a(1-i)}{\delta\sqrt{2}}\Big{)}\Big{|}$
$\displaystyle\hskip
56.9055pt+\frac{\|g^{\prime}\|_{\Delta}+\|gk^{\prime}\|_{\Delta}|z|}{\sqrt{2}}\lim\limits_{\varepsilon\rightarrow
0^{+}}\int_{0}^{a-\varepsilon}\Big{|}\operatorname{erfc}\Big{(}\frac{u_{\varepsilon}(s)-ia}{\delta\sqrt{2}}\Big{)}\Big{|}ds\bigg{)}$
$\displaystyle\leq
e^{(\|k\|_{\Delta}+1)|z|}\bigg{(}\frac{\|g\|_{\Delta}}{2}\Big{|}\operatorname{erfc}\Big{(}\frac{a(1-i)}{\delta\sqrt{2}}\Big{)}\Big{|}$
$\displaystyle\hskip
71.13188pt+\frac{\|g^{\prime}\|_{\Delta}+\|gk^{\prime}\|_{\Delta}}{\sqrt{2}}\int_{0}^{a}\Big{|}\operatorname{erfc}\Big{(}\frac{s(1-i)}{\delta\sqrt{2}}\Big{)}\Big{|}ds\bigg{)},$
(2.8)

where in the second inequality we performed the limit $\varepsilon\rightarrow
0^{+}$ and used the estimate $|z|\leq e^{|z|}$ to get rid of the
$z$-dependency inside the brackets. Using the representation
$\operatorname{erfc}(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}e^{-(t+z)^{2}}dt$
of the complementary error function, see [1, Eq. (7.1.2)], it can be estimated
by

$\Big{|}\operatorname{erfc}\Big{(}\frac{s(1-i)}{\delta\sqrt{2}}\Big{)}\Big{|}=\frac{2}{\sqrt{\pi}}\Big{|}\int_{0}^{\infty}e^{-(t+\frac{s(1-i)}{\delta\sqrt{2}})^{2}}dt\Big{|}\leq\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}e^{-t^{2}-\frac{\sqrt{2}\,ts}{\delta}}dt\leq\min\Big{\\{}\frac{\delta\sqrt{2}}{s\sqrt{\pi}},1\Big{\\}},$

where in the last inequality we estimated either $e^{-t^{2}}\leq 1$ or
$e^{-\frac{\sqrt{2}\,ts}{\delta}}\leq 1$ to get the minimum as an upper bound.
This inequality on the one hand gives an $\delta$-independent bound of the
last integral in (2.8) as well as for every $s\in(0,a]$ the pointwise
convergence as $\delta\rightarrow 0^{+}$. With the dominated convergence
theorem this then leads to

$\displaystyle\Big{|}F_{\delta}^{(3)}(z)-\frac{g(ia)}{2}e^{ik(ia)z}\Big{|}e^{-(\|k\|_{\Delta}+1)|z|}\leq\,$
$\displaystyle\frac{\|g\|_{\Delta}}{2}\Big{|}\operatorname{erfc}\Big{(}\frac{a(1-i)}{\delta\sqrt{2}}\Big{)}\Big{|}$
$\displaystyle+\frac{\|g^{\prime}\|_{\Delta}+\|gk^{\prime}\|_{\Delta}}{\sqrt{2}}\int_{0}^{a}\Big{|}\operatorname{erfc}\Big{(}\frac{s(1-i)}{\delta\sqrt{2}}\Big{)}\Big{|}ds\overset{\delta\rightarrow
0^{+}}{\longrightarrow}0.$

Since the right hand side is independent of $|z|$ we conclude the convergence

$\lim\limits_{\delta\rightarrow
0^{+}}F_{\delta}^{(3)}(z)=\frac{g(ia)}{2}e^{ik(ia)z}\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}).$

For the same reason also

$\lim\limits_{\delta\rightarrow
0^{+}}F_{\delta}^{(2)}(z)=\frac{g(ia)}{2}e^{ik(ia)z}\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}).$

Summing up, we have proved that all the terms in (2.6) converge in
$\mathcal{A}_{1}(\mathbb{C})$ and hence (2.4) follows. ∎

The next corollary puts the result of Theorem 2.1 into the perspective of
superoscillations and proves that under certain assumptions on the frequency
function $k$, the resulting functions $F_{\delta}$ satisfy Definition 1.2.

######  Corollary 2.2.

Let $b\geq a>0$ and $k,g$ be as in Theorem 2.1. Assume, in addition, that
there exists some $k_{0}>0$ such that

$g(ia)=1,\quad k(ia)\in\mathbb{R}\setminus[-k_{0},k_{0}]\quad\text{and}\quad
k(u)\in[-k_{0},k_{0}]\text{ for every }u\in\mathbb{R}.$ (2.9)

Then the functions $F_{\delta}$ in (2.3) are superoscillating with limit

$\lim\limits_{\delta\rightarrow 0^{+}}F_{\delta}(z)=e^{ik(ia)z}\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}).$ (2.10)

###### Proof.

Since $g(ia)=1$ by assumption, the convergence (2.10) follows from Theorem
2.1. Hence it remains to verify the integral representation (1.11). For this
we first define the complex Borel measure

$\sigma_{\delta}(A)\coloneqq\frac{1}{\delta\sqrt{2\pi}}\int_{A}g(u)e^{-\frac{(u-ia)^{2}}{2\delta^{2}}}du$

for Borel sets $A\subseteq\mathbb{R}$. With this measure the function
$F_{\delta}$ admits the representation

$F_{\delta}(z)=\int_{\mathbb{R}}e^{ik(u)z}d\sigma_{\delta}(u),\quad
z\in\mathbb{C}.$

In a second step we consider the Borel measure

$\mu_{\delta}(B)\coloneqq\sigma_{\delta}\big{(}\Set{u\in\mathbb{R}}{k(u)\in
B}\big{)}$

for Borel sets $B\subseteq[-k_{0},k_{0}]$ and rewrite $F_{\delta}$ as

$F_{\delta}(z)=\int_{-k_{0}}^{k_{0}}e^{ikz}d\mu_{\delta}(k),\quad
z\in\mathbb{C},$

which is exactly the form (1.11). Since by assumption $k(u)\in[-k_{0},k_{0}]$
for every $u\in\mathbb{R}$ and since the frequency $k(ia)$ of the limit
function in (2.10) lies in $\mathbb{R}\setminus[-k_{0},k_{0}]$, the functions
$F_{\delta}$ are indeed superoscillating in the sense of Definition 1.2. ∎

######  Remark 2.3.

Note that in Theorem 2.1 and Corollary 2.2 we may modify the functions $k,g$
on $\mathbb{R}\setminus[-b,b]$ as long as (2.2) and (2.9) remain valid. Since
this only changes the functions $F_{\delta}$ but not the limit
$g(ia)e^{ik(ia)z}$ in (2.4), any such modification leads to a new family of
superoscillating functions.

To illustrate our result we provide some possible choices of functions $k$ and
$g$ which lead to superoscillations in Corollary 2.2. In particular, we will
use the functions $k$ from the original paper [17].

######  Example 2.4.

Possible choices of the frequency function are

$k_{1}(u)=\frac{1}{1+\frac{u^{2}}{2}},\quad k_{2}(u)=\frac{1}{\cosh(u)},\quad
k_{3}(u)=e^{-\frac{u^{2}}{2}},\quad k_{4}(u)=\cos(u).$

All these functions satisfy $k_{j}(u)\in[-1,1]$ for $j=1,2,3,4$ and for every
$u\in\mathbb{R}$. Evaluated for complex arguments these functions admit the
values

$k_{1}(ia)=\frac{1}{1-\frac{a^{2}}{2}},\quad k_{2}(ia)=\frac{1}{\cos(a)},\quad
k_{3}(ia)=e^{\frac{a^{2}}{2}},\quad k_{4}(ia)=\cosh(a),$

and hence $k_{1}(ia)>1$ for every $a\in(0,\sqrt{2})$, $k_{2}(ia)>1$ for every
$a\in(0,\frac{\pi}{2})$, and $k_{3,4}(ia)>1$ for every $a>0$. With the
constant $g_{j}\equiv 1$ for $j=1,2,3,4$, the assumptions of Corollary 2.2 are
satisfied and we end up with superoscillating functions $F_{\delta}$. Another
possible choice considered in [17] is for any $0<a\leq 2$ the functions

$g_{5}(u)=\begin{cases}1,&\text{if }u\in\overline{\Delta},\\\ 0,&\text{if
}u\in\mathbb{R}\setminus[-a,a],\end{cases}\quad\text{and}\quad
k_{5}(u)=\begin{cases}1-\frac{u^{2}}{2},&\text{if }u\in\overline{\Delta}\\\
0,&\text{if }u\in\mathbb{R}\setminus[-a,a].\end{cases}$

Here, $\Delta$ is the triangle from Theorem 2.1 with $b=a$. Also these
functions satisfy the assumtions of Corollary 2.2 and lead to superoscillating
functions $F_{\delta}$.

##  3\. Superoscillating sinc-functions

In [18] M.V. Berry considered another type of superoscillating functions

$F_{\delta}(z)=\frac{2}{\delta}e^{-\frac{1}{\delta}}\operatorname{sinc}\bigg{(}\sqrt{z^{2}-\frac{2iaz}{\delta}-\frac{1}{\delta^{2}}}\bigg{)},\quad
z\in\mathbb{C},$ (3.1)

for some fixed $a>1$ and every $\delta>0$; here
$\operatorname{sinc}(z)\coloneqq\frac{\sin(z)}{z}$ denotes the Sinus
cardinalis and the complex square root is fixed by

$0\leq\operatorname{Arg}(\sqrt{\,\cdot\,}\,)<\pi.$ (3.2)

The aim of this section is to verify that (3.1) is indeed superoscillatory in
the sense of Definition 1.2. This will be done in two steps. In Theorem 3.2 we
prove the convergence (1.12) and with the help of Lemma A.2 we conclude the
integral representation (1.11).

We start with a technical estimate of the complex square root, which will be
important for the $\mathcal{A}_{1}$-convergence of the functions (3.1) in
Theorem 3.2.

######  Lemma 3.1.

For any $a>1$ there exists some $C\geq 0$, such that

$\big{|}\sqrt{z^{2}-2iaz-1}+az-i\big{|}\leq C\min\\{|z|,|z|^{2}\\},\quad
z\in\mathbb{C}.$ (3.3)

###### Proof.

In the first step we consider the function
$f(z)\coloneqq\sqrt{z^{2}-2iaz-1}-az+i$. Assume that $f(z_{0})=0$ for some
$z_{0}\in\mathbb{C}$, that is,

$\sqrt{z_{0}^{2}-2iaz_{0}-1}=az_{0}-i.$

Squaring both sides leads to $z_{0}^{2}=a^{2}z_{0}^{2}$ and since obviously
$z_{0}\neq 0$ by the choice of the square root in (3.2), this is a
contradiction to $a>1$. Hence $f$ does not have any zeros. Since $a>1$ by
assumption we also conclude

$\lim\limits_{|z|\rightarrow\infty}|f(z)|=\infty,$

and consequently there exists $c>0$ such that

$|f(z)|\geq c,\quad z\in\mathbb{C}.$ (3.4)

In the second step we consider $g(z)\coloneqq\sqrt{z^{2}-2iaz-1}+az-i$. Using
(3.4), this function can be estimated by

$|g(z)|=\Big{|}\frac{g(z)f(z)}{f(z)}\Big{|}=\Big{|}\frac{(1-a^{2})z^{2}}{f(z)}\Big{|}\leq\frac{(a^{2}-1)|z|^{2}}{c},\quad
z\in\mathbb{C}.$ (3.5)

Moreover, $g$ can also be estimated by

$|g(z)|\leq\sqrt{(|z|+a)^{2}}+a|z|+1=(1+a)(1+|z|),\quad z\in\mathbb{C}.$ (3.6)

Using now (3.5) for $|z|\leq 1$ and (3.6) for $|z|\geq 1$ this implies the
estimate (3.3). ∎

######  Theorem 3.2.

For every $a>1$ and $\delta>0$ the functions $F_{\delta}$ in (3.1) belong to
the space $\mathcal{A}_{1}(\mathbb{C})$ and converge as

$\lim\limits_{\delta\rightarrow 0^{+}}F_{\delta}(z)=e^{iaz}\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}).$ (3.7)

###### Proof.

For this proof it will be convenient to use the notation

$R_{\delta}(z)\coloneqq\sqrt{z^{2}-\frac{2iaz}{\delta}-\frac{1}{\delta^{2}}},\quad
z\in\mathbb{C},$ (3.8)

so that we can write
$F_{\delta}(z)=\frac{2}{\delta}e^{-\frac{1}{\delta}}\operatorname{sinc}(R_{\delta}(z))$.
Although the complex square root in (3.8) is not entire, in the series
expansion of the $\operatorname{sinc}$-function only even powers appear and we
still end up with the entire function

$F_{\delta}(z)=\frac{2}{\delta}e^{-\frac{1}{\delta}}\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}R_{\delta}(z)^{2n}=\frac{2}{\delta}e^{-\frac{1}{\delta}}\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}\Big{(}z^{2}-\frac{2iaz}{\delta}-\frac{1}{\delta^{2}}\Big{)}^{n}.$

For $\delta>0$ fixed and $z\in\mathbb{C}$ sufficiently large we have
$1\leq|R_{\delta}(z)|\leq 2|z|$ and hence the exponential bound

$|F_{\delta}(z)|=\frac{2}{\delta}e^{-\frac{1}{\delta}}\frac{|\sin(R_{\delta}(z))|}{|R_{\delta}(z)|}\leq\frac{1}{\delta}e^{-\frac{1}{\delta}}\big{|}e^{iR_{\delta}(z)}-e^{-iR_{\delta}(z)}\big{|}\leq\frac{2}{\delta}e^{-\frac{1}{\delta}}e^{2|z|}$

holds for all $z\in\mathbb{C}$ sufficiently large. It follows that
$F_{\delta}\in\mathcal{A}_{1}(\mathbb{C})$.

In order to show that $F_{\delta}$ converges as in (3.7) we first estimate the
difference between $F_{\delta}(z)$ and
$G_{\delta}(z)\coloneqq\frac{2}{\delta}e^{-\frac{1}{\delta}}\operatorname{sinc}(\frac{i}{\delta}-az)$.
Using the identity
$\operatorname{sinc}(\xi)=\frac{1}{2}\int_{-1}^{1}e^{it\xi}dt$,
$\xi\in\mathbb{C}$, we find

$F_{\delta}(z)-G_{\delta}(z)=\frac{1}{\delta}e^{-\frac{1}{\delta}}\int_{-1}^{1}\big{(}e^{itR_{\delta}(z)}-e^{it(\frac{i}{\delta}-az)}\big{)}dt,$

and with the help of (A.1) we obtain

$\begin{split}|F_{\delta}(z)-G_{\delta}(z)|&\leq\frac{1}{\delta}e^{-\frac{1}{\delta}}\int_{-1}^{1}|t|\Big{|}R_{\delta}(z)+az-\frac{i}{\delta}\Big{|}e^{|t|\max\\{|R_{\delta}(z)|,|\frac{i}{\delta}-az|\\}}dt\\\
&\leq\frac{1}{\delta}e^{-\frac{1}{\delta}}\int_{-1}^{1}\Big{|}R_{\delta}(z)+az-\frac{i}{\delta}\Big{|}e^{|t|(|R_{\delta}(z)+az-\frac{i}{\delta}|+|az-\frac{i}{\delta}|)}dt.\end{split}$
(3.9)

Since the estimate (3.3) translates into

$\Big{|}R_{\delta}(z)+az-\frac{i}{\delta}\Big{|}=\frac{1}{\delta}\big{|}\sqrt{\delta^{2}z^{2}-2ia\delta
z-1}+a\delta z-i\big{|}\leq C\min\\{|z|,\delta|z|^{2}\\},$

we can further estimate (3.9) in the form

$\begin{split}|F_{\delta}(z)-G_{\delta}(z)|&\leq
C|z|^{2}e^{-\frac{1}{\delta}}\int_{-1}^{1}e^{|t|(C|z|+a|z|+\frac{1}{\delta})}dt\\\
&\leq
C|z|^{2}e^{(C+a)|z|}e^{-\frac{1}{\delta}}\int_{-1}^{1}e^{\frac{|t|}{\delta}}dt\\\
&=2C\delta|z|^{2}e^{(C+a)|z|}e^{-\frac{1}{\delta}}(e^{\frac{1}{\delta}}-1)\\\
&\leq 2C\delta|z|^{2}e^{(C+a)|z|};\end{split}$ (3.10)

in the last inequality we used $|z|^{2}\leq 2e^{|z|}$, which follows from the
power series representation of the exponential.

In a second step we estimate the difference between $G_{\delta}(z)$ and
$e^{iaz}$. For this, we again use
$\operatorname{sinc}(\xi)=\frac{1}{2}\int_{-1}^{1}e^{it\xi}dt$,
$\xi\in\mathbb{C}$, as well as integration by parts to rewrite $G_{\delta}$ as

$\displaystyle G_{\delta}(z)$
$\displaystyle=\frac{1}{\delta}e^{-\frac{1}{\delta}}\int_{-1}^{1}e^{it(\frac{i}{\delta}-az)}dt$
$\displaystyle=-e^{-\frac{1}{\delta}}\int_{-1}^{1}\Big{(}\frac{d}{dt}e^{-\frac{t}{\delta}}\Big{)}e^{-iatz}dt$
$\displaystyle=e^{iaz}-e^{-\frac{2}{\delta}}e^{-iaz}-iaze^{-\frac{1}{\delta}}\int_{-1}^{1}e^{-\frac{t}{\delta}}e^{-iatz}dt.$

This representation allows the estimate

$\begin{split}|G_{\delta}(z)-e^{iaz}|&\leq\Big{(}e^{-\frac{2}{\delta}}+a|z|e^{-\frac{1}{\delta}}\int_{-1}^{1}e^{-\frac{t}{\delta}}dt\Big{)}e^{a|z|}\\\
&=\big{(}e^{-\frac{2}{\delta}}+a\delta|z|(1-e^{-\frac{2}{\delta}})\big{)}e^{a|z|}\\\
&\leq(e^{-\frac{2}{\delta}}+a\delta|z|)e^{a|z|}\\\
&\leq(e^{-\frac{2}{\delta}}+a\delta)e^{(a+1)|z|},\end{split}$ (3.11)

where in the last inequality we used $|z|\leq e^{|z|}$, which follows from the
power series representation of the exponential.

Combining now (3.10) and (3.11) leads to the estimate

$|F_{\delta}(z)-e^{iaz}|e^{-(C+a+1)|z|}\leq
4C\delta+e^{-\frac{2}{\delta}}+a\delta,\quad z\in\mathbb{C},$

and since the right hand side is $z$-independent and converges to zero as
$\delta\rightarrow 0^{+}$, this shows $\lim_{\delta\rightarrow
0^{+}}F_{\delta}(z)=e^{iaz}$ in $\mathcal{A}_{1}(\mathbb{C})$. ∎

The convergence result of Theorem 3.2 together with the integral
representation of the $\operatorname{sinc}$-function in Lemma A.2 imply the
superoscillatory property of the functions $F_{\delta}$ in (3.1).

######  Corollary 3.3.

For every $a>1$ the functions $F_{\delta}$ in (3.1) are superoscillating with
limit

$\lim\limits_{\delta\rightarrow 0^{+}}F_{\delta}(z)=e^{iaz}\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}).$ (3.12)

###### Proof.

The convergence (3.12) was shown in Theorem 3.2 and it remains the integral
representation (1.11). From (A.3) with $z$ and $b$ replaced by
$z-\frac{ia}{\delta}$ and $\frac{\sqrt{a^{2}-1}}{\delta}$, respectively, we
obtain

$F_{\delta}(z)=\frac{2}{\delta}e^{-\frac{1}{\delta}}\operatorname{sinc}\bigg{(}\sqrt{z^{2}-\frac{2iaz}{\delta}-\frac{1}{\delta^{2}}}\bigg{)}=\frac{1}{\delta}\int_{-1}^{1}e^{ikz}e^{\frac{ak-1}{\delta}}J_{0}\bigg{(}\frac{\sqrt{(a^{2}-1)(1-k^{2})}}{\delta}\bigg{)}dk.$

Using the complex Borel measure

$\mu_{\delta}(B)\coloneqq\frac{1}{\delta}\int_{B}e^{\frac{ak-1}{\delta}}J_{0}\bigg{(}\frac{\sqrt{(a^{2}-1)(1-k^{2})}}{\delta}\bigg{)}dk$

for Borel sets $B\subseteq[-1,1]$ we conclude that $F_{\delta}$ admits the
integral representation (1.11) with $k_{0}=1$ and hence the functions
$F_{\delta}$ are superoscillating according to Definition 1.2. ∎

##  4\. Superoscillations with prescribed frequencies

The standard example (1.1) of a superoscillating function is a linear
combination of plane waves

$F_{n}(z)=\sum\limits_{j=0}^{n}C_{j}(n)e^{ik_{j}(n)z},\quad z\in\mathbb{C},$
(4.1)

with frequencies $k_{j}(n)=1-\frac{2j}{n}$ and coefficients
$C_{j}(n)=\binom{n}{j}(\frac{1+k}{2})^{n-j}(\frac{1-k}{2})^{j}$. For a long
time it was not clear how many of such superoscillating functions exist. In
the recent paper [6] for any set of given frequencies $k_{j}(n)\in[-1,1]$ and
any target frequency $a\in\mathbb{R}\setminus[-1,1]$ coefficients $C_{j}(n)$
were constructed, such that the corresponding sequence (4.1) approximates the
exponential $e^{iax}$ in the sense that the Taylor series of $F_{n}(x)$ and
$e^{iax}$ coincide up to order $n$. This approximation suggests a certain
superoscillatory behaviour, but is not sufficient to conclude the convergence
$F_{n}(z)\rightarrow e^{iaz}$ in $\mathcal{A}_{1}(\mathbb{C})$. The analysis
in this section is partly inspired by these earlier considerations in [6]. In
particular, in Corollary 4.2 we conclude the $\mathcal{A}_{1}$-convergence
under sufficient conditions on the frequencies $k_{j}(n)$.

Starting from a more general perspective we shall study functions of the form

$F_{n}(z)=\sum\limits_{j=0}^{n}C_{j}(n)f_{j,n}(z),\quad z\in\mathbb{C},$ (4.2)

with frequency functions

$f_{j,n}(z)=\int_{-k_{0}}^{k_{0}}e^{ikz}d\mu_{j,n}(k),\quad z\in\mathbb{C}.$
(4.3)

The idea is now to choose the coefficients $C_{j}(n)$ in such a way that the
Taylor coefficients of $F_{n}(z)$ and $e^{iaz}$ coincide up to order $n$, that
is,

$F_{n}^{(l)}(0)=(ia)^{l},\quad n\in\mathbb{N},\,l\in\\{0,\dots,n\\},$ (4.4)

or, more explicitly using (4.3) and (4.2)

$\sum\limits_{j=0}^{n}C_{j}(n)\int_{-k_{0}}^{k_{0}}(ik)^{l}d\mu_{j,n}(k)=(ia)^{l},\quad
n\in\mathbb{N},\,l\in\\{0,\dots,n\\}.$

For each $n\in\mathbb{N}$ this is a linear system of the form

$S_{n}\mathbf{c}_{n}=\mathbf{a}_{n},\quad n\in\mathbb{N},$ (4.5)

where the matrix $S_{n}\in\mathbb{C}^{(n+1)\times(n+1)}$ and the vectors
$\mathbf{c}_{n},\mathbf{a}_{n}\in\mathbb{C}^{n+1}$ are given by

$S_{n}=\begin{pmatrix}\int_{-k_{0}}^{k_{0}}k^{0}d\mu_{0,n}(k)&\dots&\int_{-k_{0}}^{k_{0}}k^{0}d\mu_{n,n}(k)\\\
\vdots&\ddots&\vdots\\\
\int_{-k_{0}}^{k_{0}}k^{n}d\mu_{0,n}(k)&\dots&\int_{-k_{0}}^{k_{0}}k^{n}d\mu_{n,n}(k)\end{pmatrix},\quad\mathbf{c}_{n}=\begin{pmatrix}C_{0}(n)\\\
\vdots\\\ C_{n}(n)\end{pmatrix},\quad\mathbf{a}_{n}=\begin{pmatrix}a^{0}\\\
\vdots\\\ a^{n}\end{pmatrix}.$ (4.6)

The following Theorem 4.1 is of abstract nature and provides conditions such
that the functions $F_{n}$ in (4.2) are superoscillating. Using point measures
$\mu_{j,n}$ in Corollary 4.2 we complete the earlier considerations in [6]. A
different situation based on the choice of absolutely continuous measures is
treated in Corollary 4.4.

######  Theorem 4.1.

Let $k_{0}>0$, $a\in\mathbb{R}\setminus[-k_{0},k_{0}]$, $\mu_{j,n}$ be complex
Borel measures on $[-k_{0},k_{0}]$, and let $\mathbf{c}_{n}$ be a solution of
the system (4.5) for $n\in\mathbb{N}$. If there exist constants
$\kappa_{1},\kappa_{2}\geq 0$ such that for every $n\in\mathbb{N}$ and
$j\in\\{0,\dots,n\\}$

$|\mu_{j,n}|([-k_{0},k_{0}])\leq\kappa_{1}^{n}\quad\text{and}\quad|C_{j}(n)|\leq\kappa_{2}^{n}$
(4.7)

hold, then the functions $F_{n}$ in (4.2) are superoscillating with limit

$\lim_{n\rightarrow\infty}F_{n}(z)=e^{iaz}\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}).$ (4.8)

###### Proof.

First of all, it is clear that $f_{j,n}\in\mathcal{A}_{1}(\mathbb{C})$ for
every $j\in\\{0,\dots,n\\}$ and hence $F_{n}\in\mathcal{A}_{1}(\mathbb{C})$ by
the arguments given below Definition 1.2. The bounds of the measures
$\mu_{j,n}$ and the coefficients $C_{j}(n)$ in (4.7) immediately lead to the
estimate

$|F_{n}(z)|\leq\sum\limits_{j=0}^{n}|C_{j}(n)|\int_{-k_{0}}^{k_{0}}e^{|k||z|}d|\mu_{j,n}|(k)\leq(n+1)(\kappa_{1}\kappa_{2})^{n}e^{k_{0}|z|},\quad
z\in\mathbb{C}.$ (4.9)

Next, we write the difference between $F_{n}(z)$ and $e^{iaz}$ as the Taylor
series

$F_{n}(z)-e^{iaz}=\sum\limits_{l=0}^{\infty}\frac{F_{n}^{(l)}(0)-(ia)^{l}}{l!}z^{l}=\sum\limits_{l=n+1}^{\infty}\frac{F_{n}^{(l)}(0)-(ia)^{l}}{l!}z^{l},\quad
z\in\mathbb{C},$ (4.10)

where in the second equality we used (4.4). For
$z\in\mathbb{C}\setminus\\{0\\}$ the Cauchy integral formula along a circle of
radius $(1+\kappa_{1}\kappa_{2})|z|$ leads to the following bounds for the
coefficients in (4.10)

$\displaystyle\frac{|F_{n}^{(l)}(0)-(ia)^{l}|}{l!}$
$\displaystyle=\bigg{|}\frac{1}{2\pi
i}\int_{|\xi|=(1+\kappa_{1}\kappa_{2})|z|}\frac{F_{n}(\xi)-e^{ia\xi}}{\xi^{l+1}}d\xi\bigg{|}$
$\displaystyle\leq\frac{1}{2\pi(1+\kappa_{1}\kappa_{2})^{l}|z|^{l}}\int_{0}^{2\pi}\Big{|}F_{n}\big{(}(1+\kappa_{1}\kappa_{2})|z|e^{i\varphi}\big{)}-e^{ia(1+\kappa_{1}\kappa_{2})|z|e^{i\varphi}}\Big{|}d\varphi$
$\displaystyle\leq\frac{1}{2\pi(1+\kappa_{1}\kappa_{2})^{l}|z|^{l}}\int_{0}^{2\pi}\Big{(}(n+1)(\kappa_{1}\kappa_{2})^{n}e^{k_{0}(1+\kappa_{1}\kappa_{2})|z|}+e^{|a|(1+\kappa_{1}\kappa_{2})|z|}\Big{)}d\varphi$
$\displaystyle\leq\frac{(n+1)(\kappa_{1}\kappa_{2})^{n}+1}{(1+\kappa_{1}\kappa_{2})^{l}|z|^{l}}e^{|a|(1+\kappa_{1}\kappa_{2})|z|},$

where we used (4.9) and $k_{0}<|a|$ in the estimate of the integrand. Plugging
this into the Taylor series (4.10) gives

$\begin{split}|F_{n}(z)-e^{iaz}|&\leq\big{(}(n+1)(\kappa_{1}\kappa_{2})^{n}+1\big{)}e^{|a|(1+\kappa_{1}\kappa_{2})|z|}\sum\limits_{l=n+1}^{\infty}\frac{1}{(1+\kappa_{1}\kappa_{2})^{l}}\\\
&=\frac{(n+1)(\kappa_{1}\kappa_{2})^{n}+1}{\kappa_{1}\kappa_{2}(1+\kappa_{1}\kappa_{2})^{n}}e^{|a|(1+\kappa_{1}\kappa_{2})|z|},\quad
z\in\mathbb{C}\setminus\\{0\\}.\end{split}$ (4.11)

Due to (4.10) this inequality obviously holds for $z=0$ and hence we conclude
the $\mathcal{A}_{1}$-convergence

$\sup\limits_{z\in\mathbb{C}}|F_{n}(z)-e^{iaz}|e^{-|a|(1+\kappa_{1}\kappa_{2})|z|}\leq\frac{(n+1)(\kappa_{1}\kappa_{2})^{n}+1}{\kappa_{1}\kappa_{2}(1+\kappa_{1}\kappa_{2})^{n}}\rightarrow
0,\quad\text{as }n\rightarrow\infty.$

Finally, the integral representation (1.11) of the functions $F_{n}$ is
satisfied by

$F_{n}(z)=\int_{-k_{0}}^{k_{0}}e^{ikz}\sum\limits_{j=0}^{n}C_{j}(n)\mu_{j,n}(k)=\int_{-k_{0}}^{k_{0}}e^{ikz}d\mu_{n}(k),\quad
z\in\mathbb{C},$

using the Borel measure
$\mu_{n}(B)\coloneqq\sum_{j=0}^{n}C_{j}(n)\mu_{j,n}(B)$ for Borel sets
$B\subseteq[-k_{0},k_{0}]$. We have shown that the functions $F_{n}$ are
superoscillating with limit (4.8). ∎

In the next corollary we return to the initial problem of this section. The
aim is to find for given $k_{j}(n)$ coefficients $C_{j}(n)$ in (4.1) such that
the resulting functions $F_{n}$ are superoscillating. Here the measures
$\mu_{j,n}$ in Theorem 4.1 are chosen as point measures and an additional
condition on the frequencies $k_{j}(n)$ is imposed.

######  Corollary 4.2.

Let $k_{0}>0$, $k_{j}(n)\in[-k_{0},k_{0}]$, and assume that

$\prod\limits_{l=0,l\neq j}^{n}|k_{l}(n)-k_{j}(n)|\geq\kappa^{n},\quad
n\in\mathbb{N},\,j\in\\{0,\dots,n\\},$ (4.12)

holds for some $\kappa>0$. For $a\in\mathbb{R}\setminus[-k_{0},k_{0}]$ define
the coefficients

$C_{j}(n)=\prod\limits_{l=0,l\neq
j}^{n}\frac{k_{l}(n)-a}{k_{l}(n)-k_{j}(n)},\quad
n\in\mathbb{N},\,j\in\\{0,\dots,n\\}.$ (4.13)

Then the functions

$F_{n}(z)\coloneqq\sum_{j=0}^{n}C_{j}(n)e^{ik_{j}(n)z},\quad z\in\mathbb{C},$
(4.14)

are superoscillating with limit $\lim_{n\rightarrow\infty}F_{n}(z)=e^{iaz}$ in
$\mathcal{A}_{1}(\mathbb{C})$.

###### Proof.

Observe first that the functions (4.14) can be written in the form (4.2) using
the point measures

$\mu_{j,n}(B)\coloneqq\begin{cases}1,&\text{if }k_{j}(n)\in B,\\\ 0,&\text{if
}k_{j}(n)\notin B,\end{cases}$

for Borel sets $B\subseteq[-k_{0},k_{0}]$. Since the first bound in (4.7) is
trivially satisfied, it remains to check that the (unique) solution $C_{j}(n)$
of (4.5) satisfies the second bound in (4.7). The explicit form of the matrix
$S_{n}$ in this particular setting is

$S_{n}=\begin{pmatrix}k_{0}(n)^{0}&\dots&k_{n}(n)^{0}\\\
\vdots&\ddots&\vdots\\\ k_{0}(n)^{n}&\dots&k_{n}(n)^{n}\end{pmatrix}.$

This is a Vandermonde matrix, which is known to be invertible whenever the
$(k_{j}(n))_{j=0}^{n}$ are pairwise disjoint. But this is obviously the case
due to the assumption (4.12). Moreover, using Cramer’s rule and the known
determinant formula for Vandermonde matrices, one can easily derive the
explicit representation (4.13) for the coefficients $C_{j}(n)$ of the
solution, see also [6, Theorem 2.2]. The assumption (4.12) now allows to
estimate these coefficients as

$|C_{j}(n)|=\prod\limits_{l=0,l\neq
j}^{n}\frac{|k_{l}(n)-a|}{|k_{l}(n)-k_{j}(n)|}\leq\prod\limits_{l=0,l\neq
j}^{n}\frac{k_{0}+|a|}{|k_{l}(n)-k_{j}(n)|}\leq\Big{(}\frac{k_{0}+|a|}{\kappa}\Big{)}^{n},$

which shows the second bound in (4.7). It follows from Theorem 4.1 that the
functions $F_{n}$ are superoscillating. ∎

######  Remark 4.3.

Note that the frequencies $k_{j}(n)=1-\frac{2j}{n}$ in (1.2) can be estimated
by

$\prod\limits_{l=0,l\neq j}^{n}|k_{l}(n)-k_{j}(n)|=\prod\limits_{l=0,l\neq
j}^{n}\frac{2|l-j|}{n}=\frac{2^{n}j!(n-j)!}{n^{n}}=\frac{2^{n}n!}{n^{n}{n\choose
j}}\geq\frac{1}{e^{n}},$

where in the last inequality we used ${n\choose j}\leq 2^{n}$ as well as
$\frac{n^{n}}{n!}\leq e^{n}$, which is a consequence of the Stirling formula.
Hence, Corollary 4.2 applies and the coefficients $C_{j}(n)$ in (4.13) lead to
superoscillating functions $F_{n}$ in (4.14). Note that these coefficients do
not coincide with those in (1.2).

Now we turn to a different situation, where the measures $\mu_{j,n}$ in
Theorem 4.1 are absolutely continuous. Here the initial idea is to replace the
exponentials $e^{ik_{j}(n)z}$ in (4.1) by the derivatives of a bandlimited
function

$f(z)=\int_{-k_{0}}^{k_{0}}e^{ikz}h(k)dk,\quad z\in\mathbb{C},$ (4.15)

with $h\in L^{1}([-k_{0},k_{0}])$; note that $h$ is the compactly supported
Fourier transform of $f$.

######  Corollary 4.4.

Let $k_{0}>0$ and let $h\in L^{1}([-k_{0},k_{0}])$ be a nonnegative function
from the Szegö class, i.e. for some $\alpha,\beta\in[-k_{0},k_{0}]$ with
$\alpha<\beta$ one has

$\int_{\alpha}^{\beta}\frac{\ln(h(k))}{\sqrt{(\beta-k)(k-\alpha)}}dk>-\infty,$
(4.16)

and let $f$ be as in (4.15). Then, for every
$a\in\mathbb{R}\setminus[-k_{0},k_{0}]$ there exist coefficients
$C_{j}(n)\in\mathbb{C}$ such that the functions

$F_{n}(z)=\sum_{j=0}^{n}C_{j}(n)f^{(j)}(z),\quad z\in\mathbb{C},$ (4.17)

are superoscillating with limit $\lim_{n\rightarrow\infty}F_{n}(z)=e^{iaz}$ in
$\mathcal{A}_{1}(\mathbb{C})$.

###### Proof.

Note that the functions $F_{n}$ in (4.17) are of the form (4.2) if we use the
measures

$\mu_{j,n}(B)=\int_{B}(ik)^{j}h(k)dk$ (4.18)

for Borel sets $B\subseteq[-k_{0},k_{0}]$. Hence we are in the situation to
apply Theorem 4.1 but have to check the two estimates (4.7). The bound of the
absolute variations of the measures is clearly satisfied by

$|\mu_{j,n}|([-k_{0},k_{0}])=\int_{-k_{0}}^{k_{0}}|k|^{j}|h(k)|dk\leq
k_{0}^{j}\int_{-k_{0}}^{k_{0}}|h(k)|dk\leq\max\\{k_{0},1\\}^{n}\|h\|_{L^{1}}.$

To calculate the coefficients $C_{j}(n)$ we have to solve the system (4.5) or,
equivalently, the system $M_{n}\mathbf{d}_{n}=\mathbf{a}_{n}$ with the matrix
and the vectors

$(M_{n})_{j,l}=\int_{-k_{0}}^{k_{0}}k^{j+l}h(k)dk,\quad(\mathbf{d}_{n})_{j}=i^{j}C_{j}(n),\quad(\mathbf{a}_{n})_{l}=a^{l},$

where we shifted the powers $i^{j}$ of the imaginary unit from the coefficient
matrix into the solution vector. Now consider the matrix

$(\widetilde{M}_{n})_{j,l}\coloneqq\int_{\alpha}^{\beta}k^{j+l}h(k)dk,\quad
j,l\in\\{0,\dots,n\\}.$

It is shown in [32] that its lowest eigenvalue
$\widetilde{\lambda}_{n}=\inf_{0\neq
x\in\mathbb{C}^{n}}\frac{\langle\widetilde{M}_{n}x,x\rangle}{|x|^{2}}$ is
bounded from below by

$\widetilde{\lambda}_{n}\geq\kappa^{n},\quad n\in\mathbb{N},$

for some $\kappa>0$. Hence $\widetilde{M}_{n}$ is elliptic with bound
$\widetilde{M}_{n}\geq\kappa^{n}$ for every $n\in\mathbb{N}$. For
$x\in\mathbb{C}^{n+1}$ the estimate

$\langle
M_{n}x,x\rangle=\int_{-k_{0}}^{k_{0}}\Big{|}\sum\limits_{j=0}^{n}k^{j}x_{j}\Big{|}^{2}h(k)dk\geq\int_{\alpha}^{\beta}\Big{|}\sum\limits_{j=0}^{n}k^{j}x_{j}\Big{|}^{2}h(k)dk=\langle\widetilde{M}_{n}x,x\rangle\geq\kappa^{n}|x|^{2},$

shows that the same is true for the matrix $M_{n}$. Consequently, the inverse
matrix exists and is bounded by $\|M_{n}^{-1}\|\leq\frac{1}{\kappa^{n}}$.
Finally we can estimate the elements $C_{j}(n)$ of the solution by

$|C_{j}(n)|\leq|\mathbf{d}_{n}|=|M_{n}^{-1}\mathbf{a}_{n}|\leq\frac{1}{\kappa^{n}}|\mathbf{a}_{n}|\leq\frac{\sqrt{n+1}\max\\{a^{n},1\\}}{\kappa^{n}}.$

Hence, also the second bound in (4.7) is satisfied and the assertions follow
from Theorem 4.1. ∎

In the following example we make a special choice of the function $h$ in
(4.15) in order to find a close connection between the functions (4.17) and
the method presented in the paragraph below equation (1.8).

######  Example 4.5.

In this example we apply Corollary 4.4 with $k_{0}=1$ and the constant
function $h(k)=\frac{1}{2}$. The exact integral

$\int_{-1}^{1}\frac{\ln(\frac{1}{2})}{\sqrt{(1-k)(k+1)}}dk=-\pi\ln(2)>-\infty$

shows that the condition (4.16) is satisfied in this setting. Moreover, the
function $f$ in (4.15) is given by

$f(z)=\frac{1}{2}\int_{-1}^{1}e^{ikz}dk=\operatorname{sinc}(z),\quad
z\in\mathbb{C},$

which means that the functions $F_{n}$ in (4.17) become the sum of derivatives
of the $\operatorname{sinc}$-function

$F_{n}(z)=\sum\limits_{j=0}^{n}C_{j}(n)\operatorname{sinc}^{(j)}(z),\quad
z\in\mathbb{C}.$ (4.19)

This example is of particular importance, since these functions are closely
related to the method of P. Ferreira, A. Kempf and D. Lee in the series of
papers [24, 25, 26, 28, 29]. There functions of the form (1.8) were
considered, with coefficients $(c_{l})_{l}$ chosen in such a way that at
prescribed points $(x_{l})_{l}$ the function $F$ admits the prescribed values
$F(x_{l})=a_{l}$, $l\in\\{0,\dots,m\\}$. Furthermore, in the paper [27] this
construction is extended in the sense that also the values of the derivative
$F^{\prime}$ can be prescribed and it is then straight forward to also
prescribe the derivatives up to any order $n$, i.e.
$F^{(j)}(x_{j,l})=a_{j,l}$, $l\in\\{0,\dots,m\\}$, $j\in\\{0,\dots,n\\}$. This
then leads to a function $F$ of the form

$F(x)=\sum\limits_{l=0}^{m}c_{0,l}\operatorname{sinc}(x-x_{0,l})+\sum\limits_{l=0}^{m}c_{1,l}\operatorname{sinc}^{\prime}(x-x_{1,l})+\dots+\sum\limits_{l=0}^{m}c_{n,l}\operatorname{sinc}^{(n)}(x-x_{n,l}).$

In the special case that we choose $m=0$, the points
$x_{0,0}=x_{1,0}=\dots=x_{n,0}=0$ as the origin and the values
$a_{j,0}=(ia)^{j}$, we end up with exactly the function (4.19), namely

$F(x)=c_{0,0}\operatorname{sinc}(x)+c_{1,0}\operatorname{sinc}^{\prime}(x)+\dots+c_{n,0}\operatorname{sinc}^{(n)}(x).$

##  5\. Superoscillating solutions of the generalized Schrödinger equation

In this final section we derive two methods to construct new families of
superoscillating functions out of given ones. The underlying technique has its
origin in [7] and was revisited in [8, 9, 14]; it is based on the generalized
force free Schrödinger equation

$i\frac{\partial}{\partial t}\Psi(t,z)=-H\Big{(}-i\frac{\partial}{\partial
z}\Big{)}\Psi(t,z),\quad t,z\in\mathbb{C},$ (5.1)

where for some entire $H(z)=\sum_{l=0}^{\infty}h_{l}z^{l}$ the operator
$H(-i\frac{d}{dz})\coloneqq\sum_{l=0}^{\infty}h_{l}(-i)^{l}\frac{d^{l}}{dz^{l}}$
is defined as the corresponding infinite order differential expression. If we
now consider superoscillating initial conditions $\Psi_{n}(0,z)=F_{n}(z)$ of
the form (1.11), then the equation (5.1) admits the explicit solution

$\Psi_{n}(t,z)=\int_{-k_{0}}^{k_{0}}e^{iH(k)t}e^{ikz}d\mu_{n}(k),\quad
t,z\in\mathbb{C};$ (5.2)

cf. Lemma 5.1. One may expect that the solutions $\Psi_{n}(t,\,\cdot\,)$ are
superoscillatory also for $t\not=0$. This method is specified in Theorem 5.2
(i). Moreover, it turns out that in the point $z=0$ the functions
$\Psi_{n}(\,\cdot\,,0)$ are superoscillatory in the time variable $t$ as well,
see Theorem 5.2 (ii).

In the next lemma we collect some properties of the propagator of the
generalized Schrödinger equation (5.1).

######  Lemma 5.1.

Let $H:\mathbb{C}\rightarrow\mathbb{C}$ be an entire function. For every
$t\in\mathbb{C}$ the operator

$U_{t}\coloneqq\sum\limits_{m=0}^{\infty}\frac{(it)^{m}}{m!}H\Big{(}-i\frac{d}{dz}\Big{)}^{m},$
(5.3)

is continuous as an operator
$U_{t}:\mathcal{A}_{1}(\mathbb{C})\rightarrow\mathcal{A}_{1}(\mathbb{C})$ and
it acts on plane waves $e^{iaz}$ and functions $F_{n}(z)$ of the form (1.11)
as

$U_{t}e^{iaz}=e^{iH(a)t}e^{iaz}\quad\text{and}\quad
U_{t}F_{n}(z)=\int_{-k_{0}}^{k_{0}}e^{iH(k)t}e^{ikz}d\mu_{n}(k),$ (5.4)

respectively. Moreover, for every $F\in\mathcal{A}_{1}(\mathbb{C})$, the
function $\Psi(t,z)=U_{t}F(z)$, $t,z\in\mathbb{C}$, is a solution of the
generalized Schrödinger equation (5.1) with initial condition
$\Psi(0,z)=F(z)$.

###### Proof.

The fact that the operator $U_{t}$ is continuous in
$\mathcal{A}_{1}(\mathbb{C})$ was already shown in [15, Theorem 2.7]. To
investigate the action (5.4) of $U_{t}$ on plane waves $e^{iaz}$, we start
with the power series representation $H(z)=\sum_{l=0}^{\infty}h_{l}z^{l}$.
Then the operator $H(-i\frac{d}{dz})$ acts on plane waves $e^{iaz}$ as
multiplication

$H\Big{(}-i\frac{d}{dz}\Big{)}e^{iaz}=\sum\limits_{l=0}^{\infty}h_{l}(-i)^{l}\frac{d^{l}}{dz^{l}}e^{iaz}=\sum\limits_{l=0}^{\infty}h_{l}a^{l}e^{iaz}=H(a)e^{iaz}$

and hence the operator $U_{t}$ acts as

$U_{t}e^{iaz}=\sum\limits_{m=0}^{\infty}\frac{(it)^{m}}{m!}H\Big{(}-i\frac{d}{dz}\Big{)}^{m}e^{iaz}=\sum\limits_{m=0}^{\infty}\frac{(iH(a)t)^{m}}{m!}e^{iaz}=e^{iH(a)t}e^{iaz}.$
(5.5)

To determine the action of $U_{t}$ on functions
$F_{n}(z)=\int_{-k_{0}}^{k_{0}}e^{ikz}d\mu_{n}(k)$ we have to interchange the
complex derivatives with the integral. In the first step we use a version of
the dominated convergence theorem and obtain

$\frac{d^{l}}{dz^{l}}\int_{-k_{0}}^{k_{0}}e^{ikz}d\mu_{n}(k)=\int_{-k_{0}}^{k_{0}}\frac{d^{l}}{dz^{l}}e^{ikz}d\mu_{n}(k).$

Applying the dominated convergence theorem once more, we are also allowed to
carry the infinite sum $\sum_{l=0}^{\infty}$ of the operator
$H(-i\frac{d}{dz})$ inside the integral. This gives

$H\Big{(}-i\frac{d}{dz}\Big{)}\int_{-k_{0}}^{k_{0}}e^{ikz}d\mu_{n}(k)=\int_{-k_{0}}^{k_{0}}H\Big{(}-i\frac{d}{dz}\Big{)}e^{ikz}d\mu_{n}(k).$

Using the dominated convergence a third time for the series
$\sum_{m=0}^{\infty}$ of the operator $U_{t}$ finally gives

$U_{t}\int_{-k_{0}}^{k_{0}}e^{ikz}d\mu_{n}(k)=\int_{-k_{0}}^{k_{0}}U_{t}e^{ikz}d\mu_{n}(k)=\int_{-k_{0}}^{k_{0}}e^{iH(k)t}e^{ikz}d\mu_{n}(k),$

where in the second equation we used (5.5).

In the last part of the proof let $F\in\mathcal{A}_{1}(\mathbb{C})$ and
consider $\Psi(t,z)\coloneqq U_{t}F(z)$, $t,z\in\mathbb{C}$. Interchanging
derivatives and sums for similar reasons as above, we obtain that

$\displaystyle i\frac{\partial}{\partial t}\Psi(t,z)$
$\displaystyle=i\frac{\partial}{\partial
t}\sum\limits_{m=0}^{\infty}\frac{(it)^{m}}{m!}H\Big{(}-i\frac{d}{dz}\Big{)}^{m}F(z)$
$\displaystyle=-\sum\limits_{m=1}^{\infty}\frac{(it)^{m-1}}{(m-1)!}H\Big{(}-i\frac{d}{dz}\Big{)}^{m}F(z)$
$\displaystyle=-H\Big{(}-i\frac{d}{dz}\Big{)}\sum\limits_{m=0}^{\infty}\frac{(it)^{m}}{m!}H\Big{(}-i\frac{d}{dz}\Big{)}^{m}F(z)$
$\displaystyle=-H\Big{(}-i\frac{d}{dz}\Big{)}\Psi(t,z).$

Hence $\Psi(t,z)$ in indeed a solution of the generalized Schrödinger equation
(5.1). Since $U_{0}=\text{id}$ reduces to the identity operator, the solution
also satisfies the initial condition $\Psi(0,z)=F(z)$. ∎

The novelty of Theorem 5.2 below is that arbitrary superoscillating functions
of the form (1.11) are allowed, while in [7, 8, 9, 14] only the standard
example (1.1) was considered. Furthermore, it is shown that the resulting
functions $F_{n}^{(1)}$ and $F_{n}^{(2)}$ in (5.7) and (5.8) converge in
$\mathcal{A}_{1}(\mathbb{C})$, while only uniform convergence on compact sets
was proven in [7, 8, 9, 14].

######  Theorem 5.2.

Let $k_{0}>0$, $a\in\mathbb{R}\setminus[-k_{0},k_{0}]$, and $\mu_{n}$ be
complex Borel measures on $[-k_{0},k_{0}]$. Assume that the functions

$F_{n}(z)=\int_{-k_{0}}^{k_{0}}e^{ikz}d\mu_{n}(k),\quad z\in\mathbb{C},$ (5.6)

are superoscillating with limit $\lim_{n\rightarrow\infty}F_{n}(z)=e^{iaz}$ in
$\mathcal{A}_{1}(\mathbb{C})$. Then one can construct the following two new
families of superoscillating functions:

  1. (i)

For every entire function $H:\mathbb{C}\rightarrow\mathbb{C}$ the functions

$F_{n}^{(1)}(z)\coloneqq
e^{-H(a)}\int_{-k_{0}}^{k_{0}}e^{H(k)}e^{ikz}d\mu_{n}(k),\quad
z\in\mathbb{C},$ (5.7)

are superoscillating with $\lim_{n\rightarrow\infty}F_{n}^{(1)}(z)=e^{iaz}$ in
$\mathcal{A}_{1}(\mathbb{C})$.

  2. (ii)

For every entire function $H:\mathbb{C}\rightarrow\mathbb{C}$ with satisfies
$\operatorname{ran}\big{(}H|_{[-k_{0},k_{0}]}\big{)}\subseteq[-h_{0},h_{0}]$
and $H(a)\in\mathbb{R}\setminus[-h_{0},h_{0}]$ for some $h_{0}>0$ the sequence

$F_{n}^{(2)}(z)\coloneqq\int_{-k_{0}}^{k_{0}}e^{iH(k)z}d\mu_{n}(k),\quad
z\in\mathbb{C},$ (5.8)

is superoscillating with $\lim_{n\rightarrow\infty}F_{n}^{(2)}(z)=e^{iH(a)z}$
in $\mathcal{A}_{1}(\mathbb{C})$.

###### Proof.

For the proof of (i) we consider the operator (5.3) with $t=-i$. From the
action of this operator in (5.4) we obtain

$U_{-i}F_{n}(z)=\int_{-k_{0}}^{k_{0}}e^{H(k)}e^{ikz}d\mu_{n}(k)=e^{H(a)}F_{n}^{(1)}(z),\quad
z\in\mathbb{C}.$

Since $U_{-i}$ is continuous in $\mathcal{A}_{1}(\mathbb{C})$ due to Lemma 5.1
we conclude the convergence

$\lim\limits_{n\rightarrow\infty}F_{n}^{(1)}(z)=e^{-H(a)}\lim\limits_{n\rightarrow\infty}U_{-i}F_{n}(z)=e^{-H(a)}U_{-i}e^{iaz}=e^{iaz}\quad\text{in
}\mathcal{A}_{1}(\mathbb{C}),$ (5.9)

where the first identity in (5.4) was used in the last step. The fact that
$F_{n}^{(1)}$ is of the form (1.11) is clear using the complex Borel measure

$\sigma_{n}(B)=e^{-H(a)}\int_{B}H(k)d\mu_{n}(k)$

for Borel sets $B\subseteq[-k_{0},k_{0}]$. Hence, the functions $F_{n}^{(1)}$
are superoscillating with $\lim_{n\rightarrow\infty}F_{n}^{(1)}(z)=e^{iaz}$ in
$\mathcal{A}_{1}(\mathbb{C})$.

(ii) According to the $\mathcal{A}_{1}$-convergence (1.12), there exists some
$B\geq 0$, such that

$A_{n}\coloneqq\sup\limits_{z\in\mathbb{C}}|F_{n}(z)-e^{iaz}|e^{-B|z|}\rightarrow
0,\quad\text{as }n\rightarrow\infty.$

By the choice of the constants $A_{n}$, we obtain the estimate

$|F_{n}(z)-e^{iaz}|\leq A_{n}e^{B|z|},\quad z\in\mathbb{C}.$

In the proof of [15, Theorem 2.7] the authors obtain the estimate

$|U_{t}(F_{n}(z)-e^{iaz})|\leq
A_{n}e^{\widetilde{D}|t|}e^{\widetilde{B}|z|},\quad t,z\in\mathbb{C},$

for some $\widetilde{B},\widetilde{D}\geq 0$. For $z=0$ this estimate shows
the convergence

$\lim\limits_{n\rightarrow\infty}\sup\limits_{t\in\mathbb{C}}\big{|}U_{t}(F_{n}(z)-e^{iaz})|_{z=0}\big{|}e^{-\widetilde{D}|t|}\leq\lim\limits_{n\rightarrow\infty}A_{n}=0.$

In other words, we conclude the $\mathcal{A}_{1}$-convergence

$\lim\limits_{n\rightarrow\infty}U_{t}F_{n}(z)|_{z=0}=U_{t}e^{iaz}|_{z=0}$

in the variable $t$. Using the identities (5.4) this convergence can be
written as

$\lim\limits_{n\rightarrow\infty}F_{n}^{(2)}(t)=\lim\limits_{n\rightarrow\infty}\int_{-k_{0}}^{k_{0}}e^{iH(k)t}d\mu_{n}(k)=\lim\limits_{n\rightarrow\infty}U_{t}F_{n}(z)|_{z=0}=U_{t}e^{iaz}|_{z=0}=e^{iH(a)t}.$
(5.10)

Since
$\operatorname{ran}\big{(}H|_{[-k_{0},k_{0}]}\big{)}\subseteq[-h_{0},h_{0}]$
by assumption, we can choose the complex Borel measures

$\sigma_{n}(B)\coloneqq\mu_{n}\big{(}\Set{k\in[-k_{0},k_{0}]}{H(k)\in
B}\big{)}$

for Borel sets $B\subseteq[-h_{0},h_{0}]$ and transform $F_{n}^{(2)}$ into the
form

$F_{n}^{(2)}(t)=\int_{-h_{0}}^{h_{0}}e^{ikt}d\sigma_{n}(k),\quad
z\in\mathbb{C},$ (5.11)

which is the representation (1.11). Together with the convergence (5.10) to a
plane wave with frequency $H(a)\in\mathbb{R}\setminus[-h_{0},h_{0}]$ this
shows that the functions $F_{n}^{(2)}$ are superoscillating. ∎

## Appendix A

This appendix contains two technical results which are used in Section 2 and
Section 3. In Lemma A.1 we prove an elementary estimate for the difference of
two complex exponentials and in Lemma A.2 we derive an integral representation
of the $\operatorname{sinc}$-function.

######  Lemma A.1.

For every $z_{1},z_{2}\in\mathbb{C}$ one has

$|e^{z_{1}}-e^{z_{2}}|\leq|z_{1}-z_{2}|e^{\max\\{|z_{1}|,|z_{2}|\\}}.$ (A.1)

###### Proof.

First we write for every $z=x+iy\in\mathbb{C}$

$|1-e^{z}|^{2}=(1-\cos(y)e^{x})^{2}+\sin^{2}(y)e^{2x}=1-2\cos(y)e^{x}+e^{2x}.$

Using $\cos(y)\geq 1-\frac{y^{2}}{2}$ we can estimate this by

$|1-e^{z}|^{2}\leq(1-e^{x})^{2}+y^{2}e^{x}.$

Since $|1-e^{x}|\leq|x|e^{|x|}$, which follows directly from the power series
representation of the exponential, the above inequality becomes

$|1-e^{z}|^{2}\leq
x^{2}e^{2|x|}+y^{2}e^{x}\leq(x^{2}+y^{2})e^{2|x|}=|z|^{2}e^{2|\operatorname{Re}(z)|},\quad
z\in\mathbb{C}.$ (A.2)

Next consider $z_{1},z_{2}\in\mathbb{C}$ and assume
$\operatorname{Re}(z_{1})\leq\operatorname{Re}(z_{2})$. Then (A.2) implies

$|e^{z_{1}}-e^{z_{2}}|=e^{\operatorname{Re}(z_{1})}|1-e^{z_{2}-z_{1}}|\leq
e^{\operatorname{Re}(z_{1})}|z_{2}-z_{1}|e^{\operatorname{Re}(z_{2}-z_{1})}=|z_{2}-z_{1}|e^{\operatorname{Re}(z_{2})}\leq|z_{2}-z_{1}|e^{|z_{2}|},$

and in the same way for $\operatorname{Re}(z_{2})\leq\operatorname{Re}(z_{1})$
one obtains $|e^{z_{1}}-e^{z_{2}}|\leq|z_{1}-z_{2}|e^{|z_{1}|}$. These
estimates immediately lead to (A.1). ∎

To verify the integral representation (1.11) of the function (3.1) in the
proof of Corollary 2.2 we need the following integral representation of the
$\operatorname{sinc}$-function. A sketch of this proof is already given in
[18, Appendix C].

######  Lemma A.2.

For any $b>0$ one has

$\operatorname{sinc}\big{(}\sqrt{z^{2}+b^{2}}\big{)}=\frac{1}{2}\int_{-1}^{1}e^{ikz}J_{0}(b\sqrt{1-k^{2}})dk,\quad
z\in\mathbb{C}.$ (A.3)

where $J_{0}$ is the Bessel function of order zero.

###### Proof.

We start by deriving the so called Mehler-Sonine integral representation of
the Bessel function

$J_{0}(x)=\frac{2}{\pi}\lim\limits_{R\rightarrow\infty}\int_{0}^{R}\sin(x\cosh(t))dt,\quad
x>0.$ (A.4)

Using the Cauchy theorem we can transform for every $R>0$ the integral

$\displaystyle\int_{0}^{R}e^{ix\cosh(t)}dt$
$\displaystyle=i\int_{0}^{\frac{\pi}{2}}e^{ix\cosh(it)}dt+\int_{0}^{R}e^{ix\cosh(t+i\frac{\pi}{2})}dt-i\int_{0}^{\frac{\pi}{2}}e^{ix\cosh(R+it)}dt$
$\displaystyle=i\int_{0}^{\frac{\pi}{2}}e^{ix\cos(t)}dt+\int_{0}^{R}e^{-x\sinh(t)}dt-i\int_{0}^{\frac{\pi}{2}}e^{ix\cosh(R)\cos(t)}e^{-x\sinh(R)\sin(t)}dt.$

Performing the limit $R\rightarrow\infty$, the last integral vanishes due to
the dominated convergence theorem and we get

$\lim\limits_{R\rightarrow\infty}\int_{0}^{R}e^{ix\cosh(t)}dt=i\int_{0}^{\frac{\pi}{2}}e^{ix\cos(t)}dt+\int_{0}^{\infty}e^{-x\sinh(t)}dt.$

Using the classical integral representation of the Bessel function [1, Eq.
(9.1.18)] we get

$\displaystyle J_{0}(x)$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}\cos(x\cos(t))dt=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\cos(x\cos(t))dt$
$\displaystyle=\frac{2}{\pi}\operatorname{Im}\Big{(}\lim\limits_{R\rightarrow\infty}\int_{0}^{R}e^{ix\cosh(t)}dt\Big{)}=\frac{2}{\pi}\lim\limits_{R\rightarrow\infty}\int_{0}^{R}\sin\big{(}x\cosh(t)\big{)}dt,$

which is exactly the stated integral representation (A.4).

For the main part of the proof let us consider the function

$S(z)\coloneqq\operatorname{sinc}\big{(}\sqrt{z^{2}+b^{2}}\big{)},\quad
z\in\mathbb{C}.$

Since the restriction $S|_{\mathbb{R}}$ is square integrable, its Fourier
transform is given by the improper Riemann integral

$\mathcal{F}[S|_{\mathbb{R}}](k)=\frac{1}{\sqrt{2\pi}}\lim\limits_{R_{1},R_{2}\rightarrow\infty}\int_{-R_{1}}^{R_{2}}e^{-ikx}S(x)dx=\frac{1}{\sqrt{2\pi}}\lim\limits_{R_{1},R_{2}\rightarrow\infty}\int_{-R_{1}}^{R_{2}}\cos(kx)S(x)dx,\quad
k\in\mathbb{R},$

where the imaginary part of the integral vanishes in the limit due to the
symmetry of $S$. Starting with $|k|<1$, we substitute $x=b\sinh(t)$, to write
this integral as

$\mathcal{F}[S|_{\mathbb{R}}](k)=\frac{b}{\sqrt{2\pi}}\lim\limits_{\widetilde{R}_{1},\widetilde{R}_{2}\rightarrow\infty}\int_{-\widetilde{R}_{1}}^{\widetilde{R}_{2}}\cos\big{(}kb\sinh(t)\big{)}S\big{(}b\sinh(t)\big{)}\cosh(t)dt,$
(A.5)

where we used $\widetilde{R}_{i}\coloneqq\text{arsinh}(\frac{R_{i}}{b})$,
$i=1,2$. Using the trigonometric identity
$2\sin(u)\cos(v)=\sin(u+v)+\sin(u-v)$ we can write the integrand as

$\displaystyle 2b\cos\big{(}kb\sinh(t)\big{)}S\big{(}b\sinh(t)\big{)}\cosh(t)$
$\displaystyle=2\cos\big{(}kb\sinh(t)\big{)}\sin\big{(}b\cosh(t)\big{)}$
$\displaystyle=\sin\big{(}b\cosh(t)+kb\sinh(t)\big{)}+\sin\big{(}b\cosh(t)-kb\sinh(t)\big{)}.$

Since $|k|<1$ there exists some $t_{0}\in\mathbb{R}$ such that
$e^{2t_{0}}=\frac{1+k}{1-k}$. Then, it is easy to verify that
$\cosh(t_{0})=\frac{1}{\sqrt{1-k^{2}}}$ and
$\sinh(t_{0})=\frac{k}{\sqrt{1-k^{2}}}$ and consequently

$\cosh(t)\pm k\sinh(t)=\sqrt{1-k^{2}}\,\cosh(t\pm t_{0}).$

Using this identity we can further rewrite the integrand as

$\displaystyle
2b\cos\big{(}kb\sinh(t)\big{)}S\big{(}b\sinh(t)\big{)}\cosh(t)=$
$\displaystyle\sin\big{(}b\sqrt{1-k^{2}}\,\cosh(t+t_{0})\big{)}$
$\displaystyle+\sin\big{(}b\sqrt{1-k^{2}}\,\cosh(t-t_{0})\big{)}.$

Plugging this representation in the integral (A.5) gives

$\displaystyle\mathcal{F}[S|_{\mathbb{R}}](k)$
$\displaystyle=\frac{1}{2\sqrt{2\pi}}\lim\limits_{\widetilde{R}_{1},\widetilde{R}_{2}\rightarrow\infty}\int_{-\widetilde{R}_{1}}^{\widetilde{R}_{2}}\Big{(}\sin\big{(}b\sqrt{1-k^{2}}\,\cosh(t+t_{0})\big{)}$
$\displaystyle\hskip
142.26378pt+\sin\big{(}b\sqrt{1-k^{2}}\,\cosh(t-t_{0})\big{)}\Big{)}dt$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\lim\limits_{\widehat{R}_{1},\widehat{R}_{2}\rightarrow\infty}\int_{-\widehat{R}_{1}}^{\widehat{R}_{2}}\sin\big{(}b\sqrt{1-k^{2}}\,\cosh(t)\big{)}dt$
$\displaystyle=\frac{\sqrt{2}}{\sqrt{\pi}}\lim\limits_{\widehat{R}_{2}\rightarrow\infty}\int_{0}^{\widehat{R}_{2}}\sin\big{(}b\sqrt{1-k^{2}}\,\cosh(t)\big{)}dt$
$\displaystyle=\frac{\sqrt{\pi}}{\sqrt{2}}J_{0}\big{(}b\sqrt{1-k^{2}}\big{)},$

where in the second line we substituted $t\rightarrow t\mp t_{0}$, used
$\widehat{R}_{1}\coloneqq\widetilde{R}_{1}\mp t_{0}$,
$\widehat{R}_{2}\coloneqq\widetilde{R}_{2}\pm t_{0}$, in the two respective
integrals and added them together. Moreover, in the last equation we used the
representation (A.4).

For $k>1$ we use the Cauchy theorem to change the integration path to a
semicircle in the lower half space

$\begin{split}\mathcal{F}[S|_{\mathbb{R}}](k)&=\frac{1}{\sqrt{2\pi}}\lim\limits_{R_{1},R_{2}\rightarrow\infty}\int_{-R_{1}}^{R_{2}}e^{-ikx}S(x)dx\\\
&=\frac{1}{\sqrt{2\pi}}\lim\limits_{R\rightarrow\infty}\int_{-R}^{R}e^{-ikx}S(x)dx\\\
&=\frac{i}{\sqrt{2\pi}}\lim\limits_{R\rightarrow\infty}\int_{\pi}^{2\pi}R\,e^{-ikRe^{i\varphi}}S(Re^{i\varphi})e^{i\varphi}d\varphi.\end{split}$
(A.6)

Choosing $R>b$ and using that

$R^{2}-b^{2}\leq|R^{2}e^{2i\varphi}+b^{2}|\leq
R^{2}+b^{2},\quad\varphi\in[\pi,2\pi],$ (A.7)

the function $S$ in the integrand can be estimated by

$|S(Re^{i\varphi})|=\bigg{|}\frac{\sin(\sqrt{R^{2}e^{2i\varphi}+b^{2}})}{\sqrt{R^{2}e^{2i\varphi}+b^{2}}}\bigg{|}\leq\frac{e^{|\operatorname{Im}(\sqrt{R^{2}e^{2i\varphi}+b^{2}})|}}{\sqrt{R^{2}-b^{2}}},\quad\varphi\in[\pi,2\pi].$
(A.8)

Writing any $w\in\mathbb{C}$ as $w=|w|e^{i\operatorname{Arg}(w)}$ with
$\operatorname{Arg}(w)\in[0,2\pi)$, our choice (3.2) of the complex square
root shows $\sqrt{w}=\sqrt{|w|}\,e^{i\frac{\operatorname{Arg}(w)}{2}}$. Using
this, we rewrite the imaginary part

$\operatorname{Im}(\sqrt{w})=\sqrt{|w|}\,\sin\Big{(}\frac{\operatorname{Arg}(w)}{2}\Big{)}=\frac{\sqrt{|w|}}{\sqrt{2}}\sqrt{1-\cos(\operatorname{Arg}(w))}=\frac{1}{\sqrt{2}}\sqrt{|w|-\operatorname{Re}(w)},\quad
w\in\mathbb{C}.$

This representation and the estimate (A.7) can now be used to estimate the
exponent in (A.8) by

$2\big{|}\operatorname{Im}\big{(}\sqrt{R^{2}e^{2i\varphi}+b^{2}}\big{)}\big{|}^{2}=|R^{2}e^{2i\varphi}+b^{2}|-\operatorname{Re}(R^{2}e^{2i\varphi}+b^{2})\leq
R^{2}-R^{2}\cos(2\varphi)=2R^{2}\sin^{2}(\varphi).$

Hence we can further estimate (A.8) by

$|S(Re^{i\varphi})|\leq\frac{e^{-R\sin(\varphi)}}{\sqrt{R^{2}-b^{2}}},\quad\varphi\in[\pi,2\pi].$

Using this inequality, the Fourier transform (A.6) can be estimated by

$\displaystyle|\mathcal{F}[S|_{\mathbb{R}}](k)|$
$\displaystyle\leq\frac{1}{\sqrt{2\pi}}\lim\limits_{R\rightarrow\infty}\int_{\pi}^{2\pi}\frac{R}{\sqrt{R^{2}-b^{2}}}e^{(k-1)R\sin(\varphi)}d\varphi$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{\pi}^{2\pi}\lim\limits_{R\rightarrow\infty}\frac{R}{\sqrt{R^{2}-b^{2}}}e^{(k-1)R\sin(\varphi)}d\varphi$
$\displaystyle=0.$

For $k<-1$ we also conclude $\mathcal{F}[S|_{\mathbb{R}}](k)=0$ by the
symmetry of $S$.

Summing up, we have now shown that

$\mathcal{F}[S|_{\mathbb{R}}](k)=\begin{cases}\frac{\sqrt{\pi}}{\sqrt{2}}J_{0}(b\sqrt{1-k^{2}}),&\text{if
}|k|<1,\\\ 0,&\text{if }|k|>1.\end{cases}$ (A.9)

Applying the inverse Fourier transform to (A.9) we obtain

$S(x)=\frac{1}{2}\int_{-1}^{1}e^{ikx}J_{0}(b\sqrt{1-k^{2}})dk,\quad
x\in\mathbb{R}.$

Since $J_{0}$ is a bounded function, the right hand side extends to an entire
function when $x\in\mathbb{R}$ is replaced by $z\in\mathbb{C}$. Since the
holomorphic extension is unique it coincides with $S$ and we conclude the
stated integral representation (A.3). ∎

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Variational Flow Models: Flowing in Your Style
Kien Do, Duc Kieu, Toan Nguyen, Dang Nguyen, Hung Le, Dung
Nguyen, Thin Nguyen

Applied Artificial Intelligence Institute (A2I2), Deakin University,

{k.do, v.kieu, k.nguyen, d.nguyen, thai.le, dung.nguyen<EMAIL_ADDRESS>
§ INTRODUCTION

§ BACKGROUND

§ VARIATIONAL FLOW MODELS

§ METHOD

§ EXPERIMENTS

§ RELATED WORK

§ CONCLUSION

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