#  $\ell^{1}$-summability and Fourier series of B-splines with respect to
their knots

Martin Buhmann  Justus-Liebig University, Lehrstuhl Numerische Mathematik,
35392 Giessen, Germany<EMAIL_ADDRESS>,  Janin Jäger
Justus-Liebig University, Lehrstuhl Numerische Mathematik, 35392 Giessen,
Germany<EMAIL_ADDRESS>and  Yuan Xu  Department of
Mathematics, University of Oregon, Eugene, OR 97403–1222, USA
<EMAIL_ADDRESS>
###### Abstract.

We study the $\ell^{1}$-summability of functions in the $d$-dimensional torus
${\mathbb{T}}^{d}$ and so-called $\ell^{1}$-invariant functions. Those are
functions on the torus whose Fourier coefficients depend only on the
$\ell^{1}$-norm of their indices. Such functions are characterized as divided
differences that have $\cos{\theta}_{1},\ldots,\cos{\theta}_{d}$ as knots for
$({\theta}_{1}\,\ldots,{\theta}_{d})\in{\mathbb{T}}^{d}$. It leads us to
consider the $d$-dimensional Fourier series of univariate B-splines with
respect to its knots, which turns out to enjoy a simple bi-orthogonality that
can be used to obtain an orthogonal series of the B-spline function.

###### Key words and phrases:

Fourier series, $\ell^{1}$-invariant, B-spline function, biorthogonality,
positive definite function

###### 2010 Mathematics Subject Classification:

41A15, 42A 16, 42A32

The first author was funded by the Deutsche Forschungsgemeinschaft (DFG-German
research foundation)Projektnummer: 461449252. The third author thanks the
Alexander von Humboldt Foundation for an AvH award that supports his visit to
Justus-Liebig University, during which the work was carried out; he was
partially supported by Simons Foundation Grant #849676.

##  1\. Introduction

We consider a problem originating from the $\ell^{1}$-summability of
multivariate Fourier series. Let $f$ be a $2\pi$-periodic function in
$L^{2}({\mathbb{T}}^{d})$ and let $\hat{f}_{\alpha}$ be the Fourier
coefficient of $f$ with the multi-index
${\alpha}=({\alpha}_{1},\ldots,{\alpha}_{d})\in{\mathbb{Z}}^{d}$. Let

$S_{n}^{(1)}(f;{\theta})=\sum_{|{\alpha}|\leq
n}\hat{f}_{\alpha}\mathrm{e}^{\mathrm{i}{\alpha}\cdot{\theta}},\quad{\theta}\in{\mathbb{T}}^{d},\quad
n\in{\mathbb{N}}_{0},$

be the $n$-th $\ell^{1}$-partial sum of its Fourier series, where
$|{\alpha}|=|{\alpha}|_{1}=|{\alpha}_{1}|+\cdots+|{\alpha}_{d}|$, so that the
summation is over indices in the $\ell^{1}$-ball of radius $n$. The
$\ell^{1}$-summability has been studied in [2, 5, 8, 9, 10, 11] and it is
closely related to the summability of Fourier series in orthogonal polynomials
on the cube [12]. The Dirichlet kernel of $S_{n}^{(1)}(f)$ turns out to be a
divided difference to be defined below in the form

$D_{n}^{(1)}({\theta})=[\cos{\theta}_{1},\ldots,\cos{\theta}_{d}]G_{n,d},\qquad{\theta}\in{\mathbb{T}}^{d},$

where $G_{n,d}$ is a function of one variable as shown in [2, 12] (see (2.2)
in the next section). The divided difference can be written as an integral
with a Peano kernel; in particular, for a $(d-1)$-times differentiable
function $F:[-1,1]\to{\mathbb{C}}$,

$[\cos{\theta}_{1},\ldots,\cos{\theta}_{d}]F=\int_{-1}^{1}F^{(d-1)}(u)M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})\mathrm{d}u,$

where $u\mapsto M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})$ is the
B-spline function, which is a piecewise polynomial function in
$C^{d-2}([-1,1])$ with $\cos{\theta}_{1},\ldots,\cos{\theta}_{d}$ as its knots
(see the next section for its definition). Motivated by the
$\ell^{1}$-summability and functions defined by the above divided difference,
we call a function $\ell^{1}$-invariant if $\hat{f}_{\alpha}=\hat{f}_{\beta}$
whenever $|{\alpha}|=|{\beta}|$ and study properties of such functions.

Our analysis is partially motivated by the study in [2], where the
$\ell^{1}$-summability of the Fourier transform in ${\mathbb{R}}^{d}$, defined
by

$R_{\rho,\mathrm{d}}^{(1)}(f;x)=\int_{|v|_{1}\leq\rho}\hat{f}(v)\mathrm{e}^{\mathrm{i}v\cdot
x}\mathrm{d}v,\quad x\in{\mathbb{R}}^{d}\quad\hbox{and}\quad\rho\geq 0,$

is treated and its associated Dirichlet kernel is shown to be given as a
divided difference,

${\mathfrak{D}}_{\rho,\mathrm{d}}(x)=\left[x_{1}^{2},\ldots,x_{d}^{2}\right]{\mathfrak{G}}_{\rho,d},\qquad
x\in{\mathbb{R}}^{d},$

where ${\mathfrak{G}}_{\rho,d}$ is a function of one variable. The Fourier
transform of the B-spline function $x\mapsto
M_{d-1}(u|x_{1}^{2},\ldots,x_{d}^{2})$, considered as a function of its knots,
is analyzed in [2], which turns out to enjoy a rich structure and provides
necessary tools for studying the class of $\ell^{1}$-invariant functions
$f(\|\cdot\|_{1})$ defined on ${\mathbb{R}}^{d}$. In particular, it leads to a
characterization of $f:{\mathbb{R}}_{+}\to{\mathbb{R}}$ so that $x\mapsto
f(\|\cdot\|_{1})$ is a positive definite function on ${\mathbb{R}}^{d}$.

We will show that the $\ell^{1}$-invariant functions on the torus are all
given by divided differences with knots
$\cos{\theta}_{1},\ldots,\cos{\theta}_{d}$ and we will study the Fourier
series of the $B$-spline ${\theta}\mapsto
M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})$, which is
$\ell^{1}$-invariant. While the Fourier transform of the B-spline $x\mapsto
M_{d-1}(u|x_{1}^{2},\ldots,x_{d}^{2})$ on ${\mathbb{R}}^{d}$ satisfies an
integral recursive relation in dimension $d$, the Fourier coefficients of the
$B$-spline $x\mapsto M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})$ on
${\mathbb{T}}^{d}$ satisfy a somewhat surprising biorthogonal relation with a
family of polynomials. Let $m_{n,d}$ denote the Fourier coefficients of the
$B$-spline function with the index $|{\alpha}|=n$. Then there is a sequence of
polynomials $h_{n,d}$, given in terms of the Gegenbauer polynomials, such that
$\\{m_{n,d}:n\in{\mathbb{N}}_{0}\\}$ and $\\{h_{n,d}:n\in{\mathbb{N}}_{0}\\}$
are biorthogonal in the sense that

$\int_{-1}^{1}m_{n,d}(u)h_{\ell,d}(u)\,\mathrm{d}u=\delta_{n,\ell},\qquad
n,\ell=0,1,2,\ldots.$

This orthogonal relation can be used to derive the Fourier orthogonal series
of the B-spline function in the Gegenbauer polynomials explicitly; the first
term of the series gives, in particular, that

$\frac{1}{(2\pi)^{d}}\int_{{\mathbb{T}}^{d}}M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})\,\mathrm{d}{\theta}=\frac{\Gamma(\frac{d+1}{2})}{\sqrt{\pi}\Gamma(\frac{d}{2})(d-1)!}(1-u^{2})_{+}^{\frac{d-2}{2}},$

an identity that appears to be new. We will also give a characterization of
$\ell^{1}$-invariant functions that are either positive definite or strictly
positive definite on ${\mathbb{T}}^{d}$.

The paper is organized as follows. We recall the definition and basic
properties of $\ell^{1}$-summability in the next section and establish several
necessary identities. The Fourier orthogonal series of the B-spline with
respect to its knot is given in the third section. The positive definite
functions of $\ell^{1}$-invariant functions are discussed in the fourth
section.

##  2\. $\ell^{1}$-summability on ${\mathbb{T}}^{d}$

Let $f$ be a $2\pi$-periodic function defined on ${\mathbb{T}}^{d}$. If $f\in
L^{2}({\mathbb{T}}^{d})$, then the Fourier series of $f$ is defined by

$f(x)=\sum_{{\alpha}\in{\mathbb{Z}}^{d}}\hat{f}_{\alpha}\mathrm{e}^{\mathrm{i}{\alpha}\cdot
x},\;x\in{\mathbb{T}}^{d},\quad\hbox{with}\quad\hat{f}_{\alpha}=\frac{1}{(2\pi)^{d}}\int_{{\mathbb{T}}^{d}}f(y)\mathrm{e}^{\mathrm{i}{\alpha}\cdot
y}\,\mathrm{d}y,\;\alpha\in{\mathbb{Z}}^{d}.$

We study the class of periodic functions that we call $\ell^{1}$-invariant.

######  Definition 2.1.

A function $f:{\mathbb{T}}^{d}\to{\mathbb{R}}$ is called $\ell^{1}$-invariant
if

$\hat{f}_{\alpha}=\hat{f}_{{\beta}}\quad\hbox{whevever $|{\alpha}|=|{\beta}|$
for ${\alpha},{\beta}\in{\mathbb{Z}}^{d}$}.$

We denote the Fourier coefficient $\hat{f}_{\alpha}$ of such a function by
$\hat{f}_{|{\alpha}|}$.

If $f$ is $\ell^{1}$-invariant, then its Fourier series is of the form

(2.1) $f(x)=\sum_{n=0}^{\infty}\hat{f}_{n}E_{n}(x),\qquad
E_{n}(x)=\sum_{|{\alpha}|=n}\mathrm{e}^{\mathrm{i}{\alpha}\cdot x},\quad
x\in{\mathbb{T}}^{d},$

where $|{\alpha}|$ is the $\ell^{1}$-norm, that is
$|{\alpha}|:=|{\alpha}_{1}|+\cdots+|{\alpha}_{d}|$, of
${\alpha}\in{\mathbb{Z}}^{d}$.

A function $f$ on ${\mathbb{T}}^{d}$ is called $\ell^{1}$-summable if its
partial sum $S_{n}^{(1)}f$ over the expanding $\ell^{1}$-ball, defined by

$S_{n}^{(1)}f(x)=\sum_{|{\alpha}|\leq
n}\hat{f}_{\alpha}\mathrm{e}^{\mathrm{i}{\alpha}\cdot x},\qquad
x\in{\mathbb{T}}^{d},$

converges to $f$. The partial sum can be written as an integral operator

$S_{n}^{(1)}f(x)=\frac{1}{(2\pi)^{d}}\int_{{\mathbb{T}}^{d}}f(y)D_{n,d}(x-y)\,\mathrm{d}y=f*D_{n,d}(x),$

where the kernel $D_{n,d}$ is the analog of the Dirichlet kernel defined by

$D_{n,d}(y):=\sum_{|{\alpha}|\leq n}\mathrm{e}^{\mathrm{i}{\alpha}\cdot
y},\qquad y\in{\mathbb{T}}^{d}.$

It is shown in [2, 12] that the kernel $D_{n,d}$ can be written as a divided
difference

$D_{n,d}(x)=[\cos x_{1},\ldots,\cos x_{d}]G_{n,d},$

where $G_{n,d}$ is a univariant function defined by

(2.2)
$G_{n,d}(\cos{\theta})=(-1)^{\lfloor\frac{d-1}{2}\rfloor}2\cos\tfrac{{\theta}}{2}(\sin{\theta})^{d-2}\times\begin{cases}\cos(n+\tfrac{1}{2}){\theta}&\hbox{for
$d$ even},\\\ \sin(n+\tfrac{1}{2}){\theta}&\hbox{for $d$ odd}.\end{cases}$

We briefly recall the notion of a divided difference of a function that is at
least continuous. Let $f$ be a real or complex function on ${\mathbb{R}}$, and
let $m\in{\mathbb{N}}_{0}$. The $m$-th divided difference of $f$ at the
(pairwise distinct) knots, $x_{0},x_{1},\ldots,x_{m}$ in ${\mathbb{R}}$ is
defined inductively as

$[x_{0}]\,f=f(x_{0})\quad\hbox{and}\quad[x_{0},\ldots,x_{m}]\,f=\frac{[x_{0},\ldots,x_{m-1}]\,f-[x_{1},\ldots,x_{m}]\,f}{x_{0}-x_{m}}.$

The divided difference is a symmetric function of the knots. The knots of the
divided difference may coalesce. In particular, if all knots coalesce and if
the function is sufficiently differentiable, then the divided difference
collapses to

(2.3) $[x_{0},\ldots,x_{m}]f=\frac{f^{(m)}(x_{0})}{m!}\quad\hbox{if
$x_{0}=x_{1}=\cdots=x_{m}$}.$

Our analysis depends heavily on an integral representation of the divided
difference, for which we need the definition of B-spline. For
$x_{0}<\cdots<x_{m}$, the B-spline of order $m$ with knots
$x_{0},\ldots,x_{m}$ is defined by

${\mathbb{R}}\ni u\quad\to\quad
M_{m}(u|x_{0},\ldots,x_{m})=[x_{0},\ldots,x_{m}]\left\\{\frac{(\,\cdot\,-u)_{+}^{m-1}}{(m-1)!}\right\\}.$

The B-spline vanishes outside the interval $(x_{0},x_{n})$ and it is strictly
positive on the interval itself, and

$\int_{\mathbb{R}}M_{m}(u|x_{0},\ldots,x_{m})\;\,\mathrm{d}u=\frac{1}{m!}.$

For better reference, we state the integral representation of the divided
difference as a lemma.

######  Lemma 2.2.

Let $f:{\mathbb{R}}\to{\mathbb{C}}$ be $m$-times continuously differentiable.
Then

$[x_{0},\ldots,x_{m}]\,f=\int_{\mathbb{R}}f^{(m)}(u)M_{m}(u|x_{0},\ldots,x_{m})\,\mathrm{d}u.$

We shall also need the B-splines’ recurrence relation (Powell, 1982)

$\displaystyle M_{m+1}(u|x_{0},\ldots,x_{m},x_{m+1})\,$
$\displaystyle=\frac{(u-x_{0})M_{m}(u|x_{0},\ldots,x_{m})}{x_{m+1}-x_{0}}$
$\displaystyle+\frac{(x_{m+1}-u)M_{m}(u|x_{1},\ldots,x_{m+1})}{x_{m+1}-x_{0}}.$

We first write the function $E_{n}$ as an integral against the B-spline. We
will need the Gegenbauer polynomials, which are orthogonal polynomials with
respect to the weight function

$w_{\lambda}(t)=(1-t^{2})^{{\lambda}-\frac{1}{2}},\quad{\lambda}>-\tfrac{1}{2},$

on the interval $[-1,1]$. The Gegenbauer polynomial of degree $n$ is denoted
by $C_{n}^{\lambda}$ and normalized by
$C_{n}^{\lambda}(1)=\frac{(2{\lambda})_{n}}{n!}$. The Gegenbauer polynomials
satisfy the orthogonality

$c_{\lambda}\int_{-1}^{1}C_{n}^{\lambda}(t)C_{m}^{\lambda}(t)w_{\lambda}(t)\,\mathrm{d}t=\frac{{\lambda}}{n+{\lambda}}C_{n}^{\lambda}(1)\delta_{n,m},\qquad
n,m\in{\mathbb{N}}_{0},$

where $c_{\lambda}$ is a constant so that
$c_{\lambda}\int_{-1}^{1}w_{\lambda}(t)\,\mathrm{d}t=1$ and
$(a)_{n}=a(a+1)\cdots(a+n-1)$ denotes the Pochhammer symbol. For convenience,
we also define

$Z_{n}^{\lambda}(t):=\frac{n+{\lambda}}{{\lambda}}C_{n}^{\lambda}(t).$

The generating function of the Gegenbauer polynomials is given by

$\frac{1}{(1-2rt+r^{2})^{{\lambda}}}=\sum_{n=0}^{\infty}C_{n}^{\lambda}(t)r^{n},\qquad
0\leq r<1.$

Throughout this paper we define $C_{n}^{\lambda}(t)=0$ whenever $n<0$.

######  Lemma 2.3.

For $\theta=({\theta}_{1},\ldots,{\theta}_{d})\in{\mathbb{T}}^{d}$, the
function $E_{n}$ satisfies

(2.4) $\displaystyle E_{n}(\theta)\,$
$\displaystyle=[\cos{\theta}_{1},\ldots,\cos{\theta}_{d}]H_{n,d}$
$\displaystyle=\int_{-1}^{1}h_{n,d}(u)M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})\,\mathrm{d}u,$

where $H_{n,d}$ and $h_{n,d}$ are defined by

(2.5) $\displaystyle
H_{n,d}(\cos{\theta})=2(-1)^{\lfloor\frac{d-1}{2}\rfloor}(\sin{\theta})^{d-1}\times\begin{cases}-\sin(n{\theta})&\hbox{for
$d$ even},\\\ \cos(n{\theta})&\hbox{for $d$ odd},\end{cases}$

and $h_{n,d}$ is a polynomial of degree $n$ given by

(2.6) $\displaystyle h_{n,d}(u)\,$
$\displaystyle=(d-1)!\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}C_{n-2j}^{d}(u)$
$\displaystyle=(d-1)!\sum_{j=0}^{d-1}(-1)^{j}\binom{d-1}{j}Z_{n-2j}^{d-1}(u).$

###### Proof.

By its definition, $E_{n}({\theta})=D_{n}({\theta})-D_{n-1}({\theta})$, so
that $E_{n}({\theta})$ is a divided difference of $H_{n,d}=G_{n,d}-G_{n-1,d}$,
from which (2.4) follows readily with $h_{n,d}=H_{n,d}^{(d-1)}$. Moreover, the
identity (2.5) is an immediate consequence of (2.2) and elementary
trigonometric identities. Now, it is shown in [2] that

$g_{n,d}(t)=G_{n,d}^{(d-1)}(t)=f_{n,d}(t)+f_{n-1,d}(t),$

where $f_{n,d}$ is given in terms of the Gegenbauer polynomials by

$f_{n,d}(t)=(d-1)!\sum_{j=0}^{d-1}(-1)^{j}\binom{d-1}{j}C_{n-2j}^{d}(t).$

Using the relation [7, (4.7.29)]

$(n+{\lambda})C_{n}^{{\lambda}}(t)={\lambda}\left[C_{n}^{{\lambda}+1}(t)-C_{n-2}^{{\lambda}+1}(t)\right]$

with ${\lambda}=d-1$, we then obtain

$h_{n,d}=f_{n,d}-f_{n-2,d}=(d-1)!\sum_{j=0}^{d}(-1)^{j}\binom{d-1}{j}\left[C_{n-2j}^{d}(t)-C_{n-2j-2}^{d}(t)\right]$

which is the second expression of $h_{n,d}$ in (2.6) by recursion with
$Z_{n}^{d}$. Furthermore, the first identity in (2.6) follows from
$\binom{d-1}{j}+\binom{d-1}{j-1}=\binom{d}{j}$ and

$h_{n,d}=f_{n,d}-f_{n-2,d}=(d-1)!\sum_{j=0}^{d}(-1)^{j}\left(\binom{d-1}{j}+\binom{d-1}{j-1}\right)C_{n-2j}^{d},$

where we define for convenience $\binom{d-1}{m}=0$ if $m=-1$ or $m=d$. ∎

Let $N_{d}(n)=\\#\\{{\alpha}\in{\mathbb{N}}_{0}^{d}:|{\alpha}|=n\\}$ be the
cardinality of the set $\\{{\alpha}:|{\alpha}|=n\\}$. Then,
$N_{d}(n)=E_{n}(0)$. As a consequence of the identities (2.4) and (2.6), we
obtain

$N_{d}(n)=E_{n}(0)=\frac{h_{n,d}(1)}{(d-1)!}=\sum_{j=0}^{d}\frac{(-d)_{j}(2d)_{n-2j}}{j!(n-2j)!},$

where $(a)_{n}=a(a+1)\cdots(a+n-1)$ is the Pochhammer symbol. The last sum can
be written as a hypergeometric ${}_{3}F_{2}$ function evaluated at $1$, but
the series is not balanced so it does not have a closed-form formula. The
first values of $N(n,d)$ are given below

$N_{2}(n)=4n,\quad N_{3}(n)=2n^{2}+1,\quad N_{4}(n)=\frac{4}{9}n(n^{2}+2).$

The function $h_{n,d}$ satisfies a generating function identity which we state
as the following

######  Lemma 2.4.

Let $0\leq r<1$. Then

(2.7)
$(d-1)!\frac{(1-r^{2})^{d}}{(1-2ru+r^{2})^{d}}=\sum_{n=0}^{\infty}h_{n,d}(u)r^{n}.$

###### Proof.

By the explicit formula of $h_{n,d}$, we obtain

$\displaystyle\frac{1}{(d-1)!}\sum_{n=0}^{\infty}h_{n,d}(u)r^{n}\,$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}C_{n-2j}^{d}(u)r^{n}$
$\displaystyle=\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}\sum_{n=2j}^{\infty}C_{n-2j}^{d}(u)r^{n-2j}r^{2j}$
$\displaystyle=\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}r^{2j}\sum_{n=0}^{\infty}C_{n}^{d}(u)r^{n}$
$\displaystyle=(1-r^{2})^{d}\frac{1}{(1-2ru+r^{2})^{d}},$

where we have used the generating function of the Gegenbauer polynomials. ∎

Our next result is of interest in itself, which gives an explicit formula for
the divided difference of the function

$P_{r}(t):=\frac{1}{1-2rt+r^{2}},\quad 0\leq r<1,\quad t\in[-1,1].$

######  Proposition 2.5.

For $0\leq r<1$,

(2.8) $\displaystyle(d-1)!\int_{-1}^{1}\frac{1}{(1-2ru+r^{2})^{d}}$
$\displaystyle
M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})\,\mathrm{d}u$
$\displaystyle=\frac{1}{\prod_{i=1}^{d}(1-2r\cos{\theta}_{i}+r^{2})}.$

In particular,

(2.9)
$[\cos{\theta}_{1},\ldots,\cos{\theta}_{d}]P_{r}=\frac{(2r)^{d-1}}{\prod_{i=1}^{d}(1-2r\cos{\theta}_{i}+r^{2})}.$

###### Proof.

We start with the elementary identity

$\frac{1-r^{2}}{1-2r\cos\phi+r^{2}}=\sum_{n=0}^{\infty}r^{n}\mathrm{e}^{\mathrm{i}n\phi}.$

Reorganizing the $d$-fold product of this identity and setting
${\theta}=({\theta}_{1},\ldots,{\theta}_{d})$ as above, we obtain the
equalities

(2.10)
$\displaystyle\frac{(1-r^{2})^{d}}{\prod_{i=1}^{d}(1-2r\cos{\theta}_{i}+r^{2})}\,$
$\displaystyle=\sum_{n=0}^{\infty}r^{n}\sum_{|{\alpha}|=n}\mathrm{e}^{\mathrm{i}{\alpha}\cdot{\theta}}$
$\displaystyle=\sum_{n=0}^{\infty}r^{n}E_{n}({\theta})=\sum_{n=0}^{\infty}r^{n}[\cos{\theta}_{1},\ldots,\cos{\theta}_{d}]H_{n,d}$
$\displaystyle=\sum_{n=0}^{\infty}r^{n}\int_{-1}^{1}h_{n,d}(u)M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})\,\mathrm{d}u,$

from which the identity (2.8) follows from the generating function of
$h_{n,d}$. Now, it is easy to verify that

$P_{r}^{(d-1)}(u)=\frac{(d-1)!(2r)^{d-1}}{(1-2ru+r^{2})^{d}},$

so that the left-hand side of (2.8) can be identified with the divided
difference of $P_{r}$, which gives (2.9). ∎

##  3\. Fourier series of B-spline with respect to its knots

As a function of its knots, the B-spline function is a periodic function on
${\mathbb{T}}^{d}$,

${\mathbb{T}}^{d}\ni{\theta}\mapsto
M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})\in{\mathbb{R}},$

for each $u$, and we also define for convenience

${\mathcal{M}}_{d}({\alpha};{\theta}):=M_{d-1}(\cos\alpha|\cos{\theta}_{1},\ldots,\cos{\theta}_{d}).$

Studying this case is sufficient since for $|u|\geq 1$,
$M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})=0$ by definition, for all
${\theta}\in{\mathbb{T}}^{d}$. We first show that it is an integrable function
on ${\mathbb{T}}^{d}$.

######  Proposition 3.1.

For $u\in(-1,1)$, the function ${\theta}\mapsto
M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})$ is in
$L^{1}({\mathbb{T}}^{d})$.

###### Proof.

Let $u=\cos{\alpha}$ for $0<{\alpha}<\pi$ be fixed. Since the function
${\mathcal{M}}_{d}({\alpha};\cdot)$ is obvious even in each of its variables,
we only need to consider ${\theta}\in[0,\pi]^{d}$. Furthermore, the divided
difference is a symmetric function of its knots, the function
${\mathcal{M}}_{d}({\alpha};\cdot)$ is a symmetric function and it is
nonnegative, so we only need to show that it is an $L^{1}$ function on the
domain

$\triangle_{d}=\\{{\theta}=({\theta}_{1},\ldots,{\theta}_{d})\in{\mathbb{T}}^{d}:0\leq{\theta}_{d}\leq{\theta}_{d-1}\leq\cdots\leq{\theta}_{1}\leq\pi\\}.$

Indeed, the above consideration leads readily to

$\int_{{\mathbb{T}}^{d}}M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})\,\mathrm{d}{\theta}=2^{d}d!\int_{\triangle_{d}}M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})\,\mathrm{d}{\theta}.$

We start with the case $d=2$; the univariate case is trivial by continuity and
compact support. On the domain $\triangle_{2}$, the function is given by

${\mathcal{M}}_{2}({\alpha};{\theta})=\frac{\chi_{[{\theta}_{1},{\theta}_{2}]}({\alpha})}{\cos{\theta}_{2}-\cos{\theta}_{1}}=\begin{cases}0&{\alpha}\leq{\theta}_{2}\\\
\frac{1}{\cos{\theta}_{2}-\cos{\theta}_{1}}&{\theta}_{2}<{\alpha}<{\theta}_{1}\\\
0&{\alpha}\geq{\theta}_{1}\end{cases}.$

Hence, it follows readily that

$\displaystyle\int_{\triangle_{2}}{\mathcal{M}}_{2}({\alpha};{\theta})\,\mathrm{d}{\theta}\,$
$\displaystyle=\int_{\alpha}^{\pi}\int_{0}^{\alpha}\frac{1}{\cos{\theta}_{2}-\cos{\theta}_{1}}\,\mathrm{d}{\theta}_{2}\,\mathrm{d}{\theta}_{1}$
$\displaystyle=\int_{\alpha}^{\pi}\int_{0}^{\alpha}\frac{1}{2\sin\frac{{\theta}_{1}-{\theta}_{2}}{2}\sin\frac{{\theta}_{1}+{\theta}_{2}}{2}}\,\mathrm{d}{\theta}_{2}\,\mathrm{d}{\theta}_{1}$
$\displaystyle\leq\frac{\pi}{\min\\{\sin\frac{{\alpha}}{2},\cos\frac{{\alpha}}{2}\\}}\int_{\alpha}^{\pi}\int_{0}^{\alpha}\frac{1}{{\theta}_{1}-{\theta}_{2}}\,\mathrm{d}{\theta}_{2}\,\mathrm{d}{\theta}_{1}$

by elementary trigonometric inequalities. The last integral is evidently
bounded for $0<{\alpha}<\pi$, so that ${\mathcal{M}}_{1}({\alpha};{\theta})\in
L^{1}({\mathbb{T}}^{2})$ for $0<{\alpha}<\pi$.

For $d>2$, we use induction on $d$ and the already stated recurrence relation
from above for B-splines. Since

$\displaystyle{\mathcal{M}}_{d+1}(\alpha|{\theta})=\,$
$\displaystyle\frac{\cos\alpha-\cos{\theta}_{1}}{\cos{\theta}_{d+1}-\cos{\theta}_{1}}{\mathcal{M}}_{d}(\alpha|{\theta}_{1},\ldots,{\theta}_{d})$
$\displaystyle+\frac{\cos{\theta}_{d+1}-\cos\alpha}{\cos{\theta}_{d+1}-\cos{\theta}_{1}}{\mathcal{M}}_{d}(\alpha|{\theta}_{2},\ldots,{\theta}_{d+1})$

and ${\mathcal{M}}_{d+1}(\alpha|{\theta})=0$ if
$\alpha\not\in({\theta}_{1},{\theta}_{d+1})$, it follows that for
$\alpha\in[0,\pi]$ and $\theta\in\triangle_{d}$,

${\mathcal{M}}_{d+1}(\alpha|{\theta})\leq{\mathcal{M}}_{d}(\alpha|{\theta}_{1},\ldots,{\theta}_{d})+{\mathcal{M}}_{d}(\alpha|{\theta}_{2},\ldots,{\theta}_{d+1}).$

Consequently, the integrability of ${\mathcal{M}}_{d+1}({\alpha}|{\theta})$
follows eventually from the case $d=1$ by induction. ∎

Since ${\mathcal{M}}_{d}({\alpha};\cdot)$ is a nonnegative integrable
function, we can expand it into multiple Fourier series, which leads us to
consider the Fourier coefficients of the B-spline function as a function of
its knots. More interestingly, we consider the $\ell^{1}$-sum of its Fourier
coefficients.

######  Definition 3.2.

For $d\geq 2$ and $n\in{\mathbb{N}}_{0}$, we define

$m_{n,d}(u):=\frac{1}{(2\pi)^{d}}\int_{{\mathbb{T}}^{d}}M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})\frac{E_{n}({\theta})}{N_{d}(n)}\,\mathrm{d}{\theta}.$

By the definition of $E_{n}({\theta})$, $m_{n,d}$ is the $\ell^{1}$ mean of
the Fourier transform of the B-spline function ${\theta}\mapsto
M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})$ with respect to its
knots.

######  Theorem 3.3.

The family of functions $\\{m_{n,d}:n\in{\mathbb{N}}_{0}\\}$ and the family
$\\{h_{n,d}:n\in{\mathbb{N}}_{0}\\}$ are biorthogonal; more precisely,

(3.1)
$\int_{-1}^{1}m_{n,d}(u)h_{n^{\prime},d}(u)\,\mathrm{d}u=\delta_{n,n^{\prime}},\qquad
n,n^{\prime}\in{\mathbb{N}}_{0}.$

###### Proof.

Multiplying the first identity of (2.10) by $E_{n}({\theta})$ and integrating
over ${\theta}\in{\mathbb{T}}^{d}$, we obtain

$r^{n}=\frac{1}{(2\pi)^{d}}\int_{{\mathbb{T}}^{d}}\frac{(1-r^{2})^{d}}{\prod_{i=1}^{d}(1-2r\cos{\theta}_{i}+r^{2})}\frac{E_{n}({\theta})}{N_{d}(n)}\,\mathrm{d}{\theta}.$

Using (2.8) and exchanging the order of integrals on the right-hand side, we
obtain

$\displaystyle r^{n}$
$\displaystyle=\frac{(d-1)!}{(2\pi)^{d}}\int_{-1}^{1}\frac{(1-r^{2})^{d}}{(1-2ru+r^{2})^{d}}\int_{{\mathbb{T}}^{d}}M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})\frac{E_{n}({\theta})}{N_{d}(n)}\,\mathrm{d}{\theta}\,\mathrm{d}u$
$\displaystyle=(d-1)!\int_{-1}^{1}\frac{(1-r^{2})^{d}}{(1-2ru+r^{2})^{d}}m_{n,d}(u)\,\mathrm{d}u$
$\displaystyle=\sum_{k=0}^{\infty}\int_{-1}^{1}h_{k,d}(u)m_{n,d}(u)\,\mathrm{d}u\,r^{k},$

where the last step follows from (2.7). Since the above identity holds for
$|r|<1$, comparing the coefficients of $r^{n}$ proves (3.1) by linear
independence. ∎

Using the orthogonality, we can now derive a series expansion of $m_{n,d}$.
Let us start with $d=2$.

######  Proposition 3.4.

For $n=0,1,2,\ldots$, $m_{n,2}(u)=0$ if $|u|\geq 1$, and furthermore

(3.2)
$m_{n,2}(\cos{\alpha})=\frac{2}{\pi}\sum_{k=0}^{\infty}\frac{\sin((n+2k+1){\alpha})}{n+2k+1},\qquad
0<{\alpha}<\pi.$

###### Proof.

Let $n\geq 0$ be fixed. Since $m_{n,2}(\pm 1)=0$, we assume that $m_{n,2}(u)$
contains a factor $\sqrt{1-u^{2}}$ and takes the form

$m_{n,2}(u)=\frac{2}{\pi}\sqrt{1-u^{2}}\sum_{k=0}^{\infty}a_{k}\frac{U_{n+2k}(u)}{n+2k+1},$

where the coefficients $a_{k}$ are real numbers that will be determined by the
biorthogonality of (3.1) and $U_{n}$ are as usual the Chebyshev polynomials of
the second kind, satisfying $U_{n}=C^{1}_{n}$. Using the orthogonality of
$U_{n}$,

$\frac{2}{\pi}\int_{-1}^{1}U_{n}(t)U_{m}(t)\sqrt{1-t^{2}}\,\mathrm{d}t=\delta_{n,m},\qquad
n,m\geq 0,$

and the second explicit formula, in (2.6),

$h_{\ell,2}(u)=(\ell+1)U_{\ell}(u)-(\ell-1)U_{\ell-2}(u),\qquad\ell\geq 0,$

we see that (3.1) becomes, for $\ell=0,1,\ldots$,

$\displaystyle\delta_{\ell,n}=\int_{-1}^{1}m_{n,2}(u)h_{\ell,2}(u)\,\mathrm{d}u$
$\displaystyle=\sum_{k=0}^{\infty}a_{k}\frac{2}{\pi}\int_{-1}^{1}\frac{U_{n+2k}(u)}{n+2k+1}h_{\ell,2}(u)\sqrt{1-u^{2}}\,\mathrm{d}u.$

If $\ell$ and $n$ have different parity, then both the right-hand and the
left-hand side are zero. Assume now that $\ell$ and $n$ have the same parity.
If $\ell<n$, then the right-hand side is trivially zero by the orthogonality
of $U_{n}$. Thus, we only need to consider $\ell=n+2j$ for $j=0,1,\ldots$, for
which the identity becomes

$\delta_{j,0}=a_{j}-a_{j-1},\qquad j=0,1,2,\ldots,$

where $a_{-1}=0$, so that $a_{0}=1$ and $a_{j}=a_{j-1}$ for $j\geq 0$. Hence,
$a_{j}=1$ for $j=0,1,\ldots$. Now, setting $u=\cos{\alpha}$, then
$\sqrt{1-u^{2}}U_{n+2k}(u)=\sin((n+2k+1){\alpha})$, which gives the expression
$m_{n,d}$ in (3.2). ∎

It turns out, surprisingly, that the expression (3.2) can be written, for each
$n$, as a final sum.

######  Theorem 3.5.

For $n=0,1,2,\ldots$, and $0<{\alpha}<\pi$,

(3.3) $\displaystyle m_{2n,2}(\cos{\alpha})\,$
$\displaystyle=\frac{1}{2}-\frac{2}{\pi}\sum_{k=0}^{n-1}\frac{\sin((2k+1){\alpha})}{2k+1},$
(3.4) $\displaystyle m_{2n+1,2}(\cos{\alpha})\,$
$\displaystyle=\frac{1}{2}-\frac{{\alpha}}{\pi}-\frac{2}{\pi}\sum_{k=1}^{n}\frac{\sin(2k{\alpha})}{2k}.$

In particular, we obtain

(3.5)
$m_{0,2}(u)=\frac{1}{(2\pi)^{2}}\int_{{\mathbb{T}}^{2}}M_{1}(u|\cos{\theta}_{1},\cos{\theta}_{2})\,\mathrm{d}{\theta}_{1}\,\mathrm{d}{\theta}_{2}=\begin{cases}0&\,u\leq-1,\\\
\frac{1}{2}&\,-1<u<1,\\\ 0&\,u\geq 1.\end{cases}$

###### Proof.

Let $f_{0}$ and $f_{1}$ be odd $2\pi$ periodic functions so that their
restriction on $[0,\pi]$ are defined by

$f_{0}({\theta})=\begin{cases}0&{\theta}=0,\\\
\frac{\pi}{4}&0<{\theta}<\pi,\\\
0,&{\theta}=\pi,\end{cases}\qquad\hbox{and}\quad
f_{1}({\theta})=\begin{cases}\frac{{\theta}}{2}&0\leq{\theta}<\pi,\\\
0,&{\theta}=\pi.\end{cases}$

A quick computation shows that the Fourier series of $f_{0}$ is given by

$f_{0}({\theta})=\sum_{k=0}^{\infty}\frac{\sin(2k+1){\theta}}{2k+1},\qquad-\pi\leq{\theta}\leq\pi,$

where the convergence is pointwise. This gives immediately (3.5). Furthermore,
for $m_{2n,2}$, we obtain from (3.2)

$m_{2n,2}(\cos{\alpha})=\frac{2}{\pi}\sum_{k=n}^{\infty}\frac{\sin((2k+1){\alpha})}{(2k+1)}=\frac{2}{\pi}\left[\frac{\pi}{4}-\sum_{k=0}^{n-1}\frac{\sin((2k+1){\alpha})}{(2k+1)}\right],$

which is (3.3). Another quick computation shows that the Fourier series of
$f_{1}(\theta)$ is

$f_{1}({\theta})=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\sin(n{\theta})}{n}=\sum_{n=0}^{\infty}\frac{\sin((2n+1){\theta})}{2n+1}-\sum_{n=1}^{\infty}\frac{\sin(2n{\theta})}{2n},$

which shows in particular, together with the Fourier series of $f_{0}$,

$\sum_{n=1}^{\infty}\frac{\sin(2n{\theta})}{2n}=\frac{\pi}{4}-\frac{{\theta}}{2},\qquad
0<{\theta}<\pi.$

Thus, for $m_{2n+1,2}$, we obtain from (3.2) that

$m_{2n+1,2}(\cos{\alpha})=\frac{2}{\pi}\sum_{k=n+1}^{\infty}\frac{\sin(2k{\alpha})}{2k}=\frac{2}{\pi}\left[\frac{\pi}{4}-\frac{{\alpha}}{2}-\sum_{k=0}^{n}\frac{\sin(2k{\alpha})}{2k}\right],$

which is (3.4). ∎

######  Proposition 3.6.

For $d>2$, the function $m_{n,d}$ is of the form

(3.6)
$m_{n,d}(u)=(1-u^{2})^{d-\frac{3}{2}}\frac{c_{d-1}}{(d-1)!}\sum_{k=0}^{\infty}\frac{(d-1)_{k}}{k!}\frac{C_{n+2k}^{d-1}(u)}{C_{n+2k}^{d-1}(1)},\qquad-1\leq
u\leq 1,$

where $c_{d-1}$ is the normalization constant defined by its reciprocal

$c_{d-1}^{-1}=\int_{-1}^{1}(1-u^{2})^{d-\frac{3}{2}}\mathrm{d}u=\frac{\Gamma\left(\frac{1}{2}\right)\Gamma\left(d-\frac{1}{2}\right)}{(d-1)!}.$

###### Proof.

For $d>2$, the function $u\mapsto
M_{d-1}(u|\cos\\{\cdot\\}_{1},\ldots,\cos\\{\cdot\\}_{d})$ has support set in
$(-1,1)$ and has $d-2$ continuous derivatives, it follows that $m_{n,d}$ is a
continuous and $m_{n,d}^{(j)}(\pm 1)=0$ for $j=0,1,\ldots,d-2$. We assume that
$m_{n,d}$ has the series expansion

$m_{n,d}(u)=\frac{c_{d-1}}{(d-1)!}(1-u^{2})^{d-\frac{3}{2}}\sum_{k=0}^{\infty}a_{k}^{n}\frac{C_{n+2k}^{d-1}(u)}{C_{n+2k}^{d-1}(1)},$

where the coefficients $a_{k}^{n}$ are to be determined by the
biorthogonality, and the choice of index $n+2k$ comes from (3.1) and the
parity of $h_{n,d}$. Now, the orthogonality of the Gegenbauer polynomials is
equivalent to

(3.7)
$c_{d-1}\int_{-1}^{1}\frac{C_{n}^{d-1}(u)}{C_{n}^{d-1}(1)}Z_{m}^{d-1}(u)(1-u^{2})^{d-\frac{3}{2}}\,\mathrm{d}u=\delta_{n,m}.$

Using the second explicit formula of $h_{n,d}$ in (2.6), which shows that the
term of the highest degree in $h_{n,d}$ is $(d-1)!Z_{n}^{d-1}$, whereas the
term of the lowest degree in $m_{n,d}$ contains $C_{n}^{d-1}$, or the term
$k=0$ in the sum, the orthogonality (3.7) implies that the identity (3.1) for
$n^{\prime}=n$ becomes

$\displaystyle a_{0}^{n}=\delta_{n,n}=1,\quad\forall n\in{\mathbb{N}}_{0};$

moreover, for $n^{\prime}=n+2\ell$ and $\ell=1,2,3,\ldots$, (3.1) becomes

$\displaystyle
0=\int_{-1}^{1}m_{n,d}(u)h_{n+2\ell,d}(u)\,\mathrm{d}u=(d-1)!\sum_{j=0}^{d-1}(-1)^{j}\binom{d-1}{j}a_{\ell-j}^{n}.$

Thus, using $a_{j}^{n}=0$ for $j<0$ and recalling that $a_{0}^{n}=1$, we see
that $a_{j}^{n}$ satisfy

(3.8)
$\sum_{j=0}^{\ell}(-1)^{j}\binom{d-1}{j}a_{\ell-j}^{n}=\delta_{0,\ell},\qquad\ell=0,1,\ldots,d-1.$

It shows, in particular, that $a_{j}^{n}$ are independent of $n$ and they can
be determined recursively so that the solution is unique. It turns out that
the solution (3.8) is given explicitly by $a_{j}=(d-1)_{j}/j!$. To verify that
this is indeed the case, we use the identity

$a_{\ell-j}=\frac{(d-1)_{\ell-j}}{(\ell-j)!}=\frac{(d-1)_{\ell}(-\ell)_{j}}{\ell!(2-\ell-d)_{j}},$

which follows from $(-x)_{\ell-j}=(-x)_{\ell}(-1)^{j}(1-\ell-x)_{j}$, and
$\binom{d-1}{j}=(-1)^{j}(-d+1)_{j}/j!$ to write the right-hand side of (3.8)
as a hypergeometric function,

$\sum_{j=0}^{\ell-1}(-1)^{j}\binom{d-1}{j}a_{\ell-j}^{n}=\frac{(d-1)_{\ell}}{\ell!}{}_{2}F_{1}\left(\begin{matrix}-\ell,\,1-d\\\
2-d-\ell\end{matrix};1\right)=\frac{(d-1)_{\ell}(1-\ell)_{\ell}}{\ell!(2-d-\ell)_{\ell}},$

where the last step follows from the Chu-Vandermonde identity. Since
$(-\ell+1)_{\ell}=0$ for $\ell\in{\mathbb{N}}$, this verifies that
$a_{j}=(d-1)_{j}/j!$ is the solution of (3.8). ∎

######  Theorem 3.7.

For $d>2$ and $-1\leq u\leq 1$,

$\displaystyle m_{0,d}(u)\,$
$\displaystyle=\frac{1}{(2\pi)^{d}}\int_{{\mathbb{T}}^{d}}M_{d-1}(u|\cos{\theta}_{1},\ldots,\cos{\theta}_{d})\,\mathrm{d}{\theta}$
$\displaystyle=\frac{\Gamma(\frac{d+1}{2})}{\sqrt{\pi}\Gamma(\frac{d}{2})(d-1)!}(1-u^{2})^{\frac{d-2}{2}}.$

###### Proof.

Let $g(u)=(1-u^{2})^{-\frac{d-1}{2}}$. We compute the Fourier-Gegenbauer
coefficients $\hat{g}^{d-1}_{n}$ defined by

$\displaystyle\hat{g}^{d-1}_{n}\,$
$\displaystyle=c_{d-1}\int_{-1}^{1}g(t)C_{n}^{{d-1}}(t)(1-t^{2})^{d-\frac{3}{2}}\,\mathrm{d}t$
$\displaystyle=c_{d-1}\int_{-1}^{1}C_{n}^{{d-1}}(t)(1-t^{2})^{\frac{d-2}{2}}\,\mathrm{d}t.$

By the parity of $C_{n}^{d-1}$, $\hat{g}^{d-1}_{n}=0$ if $n$ is odd. To
compute $\hat{g}_{2n}^{d-1}$, we use the connection coefficients of Gegenbauer
polynomials [1, Theorem 7.1.4’, p. 360], which gives

$C_{2n}^{d-1}(t)=\sum_{k=0}^{n}\frac{(d-1)_{2n-k}(\frac{d-1}{2})_{k}}{(\frac{d-1}{2}+1)_{2n-k}k!}Z_{2n-2k}^{\frac{d-1}{2}}(u).$

Using the orthogonality of $Z_{m}^{\frac{d-1}{2}}$, we then conclude that

$\hat{g}^{d-1}_{n}=\frac{c_{d-1}}{c_{\frac{d-1}{2}}}\frac{(d-1)_{n}(\frac{d-1}{2})_{n}}{(\frac{d-1}{2}+1)_{n}n!}=\frac{c_{d-1}}{c_{\frac{d-1}{2}}}\frac{(d-1)_{n}}{n!}\frac{d-1}{2n+d-1}.$

(Note that $c_{\cdot}$ is also defined for a non-integral index.)
Consequently, the Fourier-Gegenbauer expansion of $g$ is given by

$g(u)=\sum_{n=0}^{\infty}\hat{g}_{n}^{d-1}\frac{C_{n}^{d-1}(u)}{h_{n}^{d-1}}=\frac{c_{d-1}}{c_{\frac{d-1}{2}}}\sum_{n=0}^{\infty}\frac{(d-1)_{n}}{n!}\frac{C_{2n}^{d-1}(u)}{C_{2n}^{d-1}(1)},$

where $h_{n}^{\lambda}$ denotes the
$L^{2}([-1,1],(1-t^{2})^{{\lambda}-\frac{1}{2}})$ norm of $C_{n}^{\lambda}(t)$
and it is equal to

$h_{n}^{\lambda}=\frac{{\lambda}}{n+{\lambda}}C_{n}^{\lambda}(1),\qquad
n=0,1,2,\ldots.$

Comparing with (3.6) with $n=0$, we see that

(3.9)
$m_{0,d}=(1-u^{2})^{d-\frac{3}{2}}\frac{c_{\frac{d-1}{2}}}{(d-1)!}g(u)=\frac{c_{\frac{d-1}{2}}}{(d-1)!}(1-u^{2})^{\frac{d-2}{2}},$

which is the stated result. ∎

In the case of $d=2$, it is easy to see that the explicit formula (3.2)
implies the relation

$m_{n,2}(u)-m_{n+2,2}(u)=\frac{2}{\pi}\frac{\sin(n+1){\alpha}}{n+1}=\frac{2}{\pi}\sqrt{1-u^{2}}\frac{U_{n}(u)}{U_{n}(1)},\quad
u=\cos{\alpha}.$

The following corollary gives a $d$-dimensional version of this recursive
relation for $m_{n,d}$.

######  Corollary 3.8.

For $d\geq 2$, $n=0,1,2,\ldots$,

(3.10)
$(d-1)!\sum_{j=0}^{d-1}(-1)^{j}\binom{d-1}{j}m_{n+2j,d}(u)=c_{d-1}(1-u^{2})^{d-\frac{3}{2}}\frac{C_{n}^{d-1}(u)}{C_{n}^{d-1}(1)}.$

###### Proof.

Using the explicit formula of $m_{n,d}$ in (3.6), we obtain

$\displaystyle(d-1)!\sum_{j=0}^{d-1}(-1)^{j}\binom{d-1}{j}m_{n+2j,d}(u)$
$\displaystyle\qquad=(1-u^{2})^{d-\frac{3}{2}}c_{d-1}\sum_{j=0}^{d-1}(-1)^{j}\binom{d-1}{j}\sum_{k=j}^{\infty}\frac{(d-1)_{k-j}}{(k-j)!}\frac{C_{n+2k}^{d-1}(u)}{C_{n+2k}^{d-1}(1)}$
$\displaystyle\qquad=(1-u^{2})^{d-\frac{3}{2}}c_{d-1}\sum_{k=0}^{\infty}\sum_{j=0}^{d-1}(-1)^{j}\binom{d-1}{j}\frac{(d-1)_{k-j}}{(k-j)!}\frac{C_{n+2k}^{d-1}(u)}{C_{n+2k}^{d-1}(1)},$

where we have used the convention that $(d-1)_{k-j}=0$ if $j>k$ and
$\binom{d-1}{j}=0$ if $j>d-1$. The stated result then follows from (3.8). ∎

In particular, the identity (3.10) shows that the finite combination of
$m_{n,d}$ in the left-hand is a polynomial of degree $n$ multiplied by
$(1-t^{2})^{d-\frac{3}{2}}$. The recursive formula can be used to determine
$m_{n,d}$ if we know the first $d-1$ elements $m_{0,d},\ldots,m_{d-1,d}$.
However, notice that the explicit expression of $m_{0,d}$ in (3.9) contains
the factor $(1-u^{2})^{\frac{d-2}{2}}$, which has a power different from the
power $d-\frac{3}{2}$ in the right-hand side of (3.10), we see that an analog
of (3.3) is unlikely to hold for $d>2$; in particular, $m_{n,d}$ will not be a
polynomial when $n$ is even for $d>2$.

##  4\. Positive definite $\ell^{1}$-invariant functions

A function $f\in C({\mathbb{T}}^{d})$ is a positive definite function (PDF) if
for every $\Xi_{N}=\\{\Theta_{1},\ldots,\Theta_{N}\\}$ of pairwise distinct
points in ${\mathbb{T}}^{d}$ and $N\in{\mathbb{N}}_{0}$, the matrix

$f[\Xi_{N}]=\left[f(\Theta_{i}-\Theta_{j})\right]_{i,j=1}^{N}$

is nonnegative definite; it is called a strictly positive definite function
(SPDF) if the matrix is positive definite. Let $\Phi({\mathbb{T}}^{d})$ denote
the set of PDFs on ${\mathbb{T}}^{d}$. By the definition of PDF, the space
$\Phi({\mathbb{T}}^{d})$ is closed under linear combination with nonnegative
coefficients; that is, if $f,g\in\Phi({\mathbb{T}}^{d})$ and $c_{i}$ and
nonnegative constants, then $c_{1}f+c_{2}g\in\Phi({\mathbb{T}}^{d})$. The PDFs
on ${\mathbb{T}}^{d}$ are characterized by the following theorem

######  Theorem 4.1.

A function $f\in C({\mathbb{T}}^{d})$ is a PDF on ${\mathbb{T}}^{d}$ if and
only if the Fourier coefficients $\hat{f}_{\alpha}$ are nonnegative for all
${\alpha}\in{\mathbb{N}}_{0}^{d}$.

One direction of the theorem follows readily from the closeness of
$\Phi({\mathbb{T}}^{d})$ and that the exponential functions
$\mathrm{e}^{\mathrm{i}{\alpha}\cdot x}\in\Phi({\mathbb{T}}^{d})$ for all
${\alpha}\in{\mathbb{N}}_{0}^{d}$, since

$\sum_{i=1}^{N}\sum_{j=1}^{N}c_{i}c_{j}\mathrm{e}^{\mathrm{i}{\alpha}\cdot(\Theta_{i}-\Theta_{j})}=\left|\sum_{i=1}^{N}c_{i}\mathrm{e}^{\mathrm{i}{\alpha}\cdot\Theta_{i}}\right|^{2}\geq
0.$

In the other direction, if $f$ is PDF on ${\mathbb{T}}^{d}$ then by the
periodicity of $f$,

$\int_{{\mathbb{T}}^{d}}f(\theta)\,\mathrm{d}\theta=\frac{1}{(2\pi)^{d}}\int_{{\mathbb{T}}^{d}}\int_{{\mathbb{T}}^{d}}f(\theta-\phi)\,\mathrm{d}\theta\,\mathrm{d}\phi.$

The right-hand side is nonnegative if $f$ is a trigonometric polynomial, as
can be seen by applying a positive cubature rule for the integral over
${\mathbb{T}}^{d}$ and using the positive definiteness of $f$. In particular,
this shows that the left-hand side integral is nonnegative. Since
$\mathrm{e}^{\mathrm{i}{\alpha}\cdot{\theta}}$ is PDF, it follows from Schur’s
theorem that

$\hat{f}_{\alpha}=\frac{1}{(2\pi)^{d}}\int_{{\mathbb{T}}^{d}}f({\theta})\mathrm{e}^{-\mathrm{i}{\alpha}\cdot{\theta}}\,\mathrm{d}{\theta}\geq
0,\qquad\forall{\alpha}\in{\mathbb{N}}_{0}^{d}.$

Recall that a function $f:{\mathbb{T}}^{d}\to{\mathbb{R}}$ is
$\ell^{1}$-invariant if $\hat{f}_{\alpha}=\hat{f}_{\beta}$ for all
$|{\alpha}|=|{\beta}|$, ${\alpha},{\beta}\in{\mathbb{N}}_{0}^{d}$. These
functions are given as follows.

######  Theorem 4.2.

A function $f\in L^{1}({\mathbb{T}}^{d})$ is $\ell^{1}$-invariant if and only
if

(4.1)
$f({\theta}_{1},\ldots,{\theta}_{d})=[\cos{\theta}_{1},\ldots,\cos{\theta}_{d}]F_{d},$

where $F_{d}:[-1,1]\to{\mathbb{R}}$ is a $(d-1)$-times differentiable function
and satisfies

(4.2)
$F_{d}(\cos{\theta})=2(-1)^{\lfloor\frac{d+1}{2}\rfloor}(\sin{\theta})^{d-1}\sum_{n=0}^{\infty}a_{n}\begin{cases}-\sin(n{\theta})&\hbox{for
$d$ even},\\\ \cos(n{\theta})&\hbox{for $d$ odd},\end{cases}$

with a sequence of real numbers $\\{a_{n}\\}_{n\geq 0}$.

###### Proof.

If $f$ is the given divided difference of $F_{d}$ given in (4.2), then

$f({\theta})=\sum_{n=0}^{\infty}a_{n}[\cos{\theta}_{1},\ldots,\cos{\theta}_{d}]H_{n,d}=\frac{1}{2}\sum_{n=0}^{\infty}a_{n}E_{n}({\theta}).$

By the definition of $E_{n}({\theta})$, it follows readily that
$\hat{f}_{\alpha}=\hat{f}_{\beta}$ if $|{\alpha}|=|{\beta}|$. Moreover, when
its knots coalesce, the divided difference becomes a derivative as we have
mentioned before. By (2.3) and the $(d-1)$-th differentiability of $F$, $f$ is
continuous.

In the other direction, if $f$ is $\ell^{1}$-invariant, then its Fourier
series is given by (2.1). By (2.4), we obtain that
$f({\theta}_{1},\ldots,{\theta}_{d})=[\cos{\theta}_{1},\ldots,\cos{\theta}_{d}]F$
with $F$ given by

$F(u)=\sum_{n=0}^{\infty}\hat{f}_{n}H_{n,d}(u),$

so that $F$ is of the form (4.2) with $a_{n}=\widehat{f}_{n}$ by (2.5). The
continuity of $f$ requires that $F$ has continuous derivatives of $(d-1)$
order by (2.3). ∎

If $f$ is $\ell^{1}$-invariant, then $\hat{f}_{\alpha}=\hat{f}_{|{\alpha}|}$,
so that $\hat{f}_{\alpha}\geq 0$ if and only if $\hat{f}_{n}\geq 0$ in (2.1).
Consequently, the following characterization of PDFs holds.

######  Theorem 4.3.

Let $f\in C({\mathbb{T}}^{d})$ be $\ell^{1}$-invariant. Then $f$ is a PDF on
${\mathbb{T}}^{d}$ if and only if $f$ is given by (4.1) and (4.2) with
$\hat{f}_{n}\geq 0$ for all $n\in{\mathbb{N}}_{0}$.

The SPDFs have been characterized in [4, Theorem 1], where it is proved that a
PDF function is also SPDF if and only if the set of indices
$\mathcal{G}=\\{\alpha\in{\mathbb{Z}}^{d}\mid\hat{f}_{\alpha}>0\\}$ intersects
all the translations of each subgroup of ${\mathbb{Z}}^{d}$ that has the form

$(a_{1}{\mathbb{Z}},a_{2}{\mathbb{Z}},\ldots,a_{d}{\mathbb{Z}}),\quad
a_{1},{\alpha}_{2},\ldots,a_{d}\in{\mathbb{N}}.$

More precisely, for every pair of vectors
$\gamma\in{\mathbb{Z}}^{d},\beta\in{\mathbb{N}}^{d}$ there exists a
$z\in{\mathbb{Z}}^{d}$ with $\hat{f}_{\alpha}>0$ and
$\alpha_{j}=\gamma_{j}+z_{j}\beta_{j}$, $j=1,\ldots,d$. For
$\ell^{1}$-invariant functions, the SPDFs are characterized below.

######  Theorem 4.4.

Let $f\in C({\mathbb{T}}^{d})$ be $\ell^{1}$-invariant. Then $f$ is a SPDF on
${\mathbb{T}}^{d}$ if and only if $f$ is given by (4.1) and (4.2), $f$ is PDF
and for any pair $n<\ell\in{\mathbb{N}}$ there is an $m\in{\mathbb{N}}$ with
$\hat{f}_{n+m\ell}>0$ or $\hat{f}_{(\ell-n)+m\ell}>0$.

###### Proof.

We prove that the given condition is equivalent to the condition given in [4],
if we assume the kernel to be $\ell^{1}$-summable.

We start with sufficiency. Assume $f$ is PDF and satisfies the condition of
the theorem. For any two vectors
$\gamma\in{\mathbb{Z}}^{d},\beta\in{\mathbb{N}}^{d}$ we can assume without
loss of generality that $0\leq\gamma_{j}\leq\beta_{j}$, $j=1,\ldots d$. We
define $n=|\gamma|$ and $\ell=|\beta|$. Then there exists an
$m\in{\mathbb{N}}_{0}$ with $\hat{f}_{n+\ell m}>0$ or $\hat{f}_{(\ell-n)+\ell
m}>0$. If $\hat{f}_{n+\ell m}>0$, then

$\hat{f}_{\gamma+m\beta}=\hat{f}_{n+\ell m}>0$

since $|\gamma+m\beta|=n+m\ell$. Whereas if $\hat{f}_{(n-\ell)+\ell m}>0$, we
can choose $m^{\prime}=m+1$. Now
$|\gamma-m^{\prime}\beta|=|\gamma-\beta-m\beta|=(\ell-n)+m\ell$ implies

$\hat{f}_{\gamma-m^{\prime}\beta}=\hat{f}_{(n-\ell)+\ell m}>0.$

For the necessity, assume the assumption of the theorem does not hold, then
there exists $\ell>n\in{\mathbb{N}}$ such that $\hat{f}_{n+m\ell}=0$ and
$\hat{f}_{(\ell-n)+\ell m}=0$ for all $m\in{\mathbb{N}}$. Set

$\gamma=(n,0,\ldots,0)\in{\mathbb{Z}}^{d},\quad\beta=(\ell,\ldots,\ell)\in{\mathbb{Z}}^{d}.$

Then for any $z\in{\mathbb{Z}}^{d}$ define $\alpha\in{\mathbb{Z}}^{d}$,
$\alpha_{j}=\gamma_{j}+z_{j}\beta_{j}$, so that with $m=|z|$,

$|\alpha|=|n+z_{1}\ell|+\ell\sum_{j=2}^{d-1}|z_{j}|=\begin{cases}n+m\ell,&z_{1}\geq
0,\\\ (\ell-n)+(m-1)\ell,&z_{1}<0.\end{cases}$

Thereby, there would be no coefficients $\hat{f}_{\alpha}>0$,
$\alpha_{j}=\gamma_{j}+z_{j}\beta_{j}$ for this choice of $\gamma$ and
$\beta$, contradicting strict positive definiteness. ∎

## References

  * [1] G. E. Andrew, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999.
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# A Method for Investigating Thermodynamics of Inaccessible Objects by
Determining the Co-Amoeba of their Partition Functions

Arijit<EMAIL_ADDRESS>Department of
Physics and NMR Research Center, Indian Institute of Science Education and
Research, Pune, India T. S<EMAIL_ADDRESS>Department of
Physics and NMR Research Center, Indian Institute of Science Education and
Research, Pune, India Mounir<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Mathematics and Physics, Xiamen University
Malaysia, 43900 Sepang, Malaysia Yen-Kheng<EMAIL_ADDRESS><EMAIL_ADDRESS>School of Mathematics and Physics, Xiamen University
Malaysia, 43900 Sepang, Malaysia

###### Abstract

Given an Ising-type system, the zeros of its complexified partition function,
known as Lee–Yang zeros, form an algebraic variety that sheds light on the
thermodynamics of that system. But in practice, we often confront with the
inaccessibility of a system for direct experimentation and then a strategic
approach is taken to investigate a probe that is interacting with the target
system. Considering such an inaccessible system, reading the probe alone, we
present a method to determine the co-amoeba, the images of the argument map of
an algebraic variety corresponding to the Lee–Yang zeros of its partition
function. In our method, starting from thermal equilibrium of target and
probe, the target can be effectively initiated at any chosen value of its
internal parameters by doing operations only on the probe. Thus, sampling the
co-amoeba corresponding to a wide of range of physical parameters is possible
which allows one to perceive the full algebraic variety and thereby
determining the thermodynamics of the system. No prior knowledge about the
internal parameters, neither any control over the target system under
investigation is required for this protocol. Finally we experimentally
demonstrate this protocol via the nuclear magnetic resonance architecture
using a system of three mutually interacting spin-1/2 nuclei.

##  1 Introduction

It is typically understood that complex numbers merely play the role of a
calculational tool in physics, since observable quantities are often
represented by real numbers. A key illustration of this is the Lee–Yang circle
theorem [1, 2], which describes the locations of the zeroes of the partition
function of the Ising model. Indeed, the theorem works by extending the
variables of the partition function into the complex plane. It states that for
an Ising chain in a magnetic field, the zeroes of the partition function,
known as _Lee–Yang zeroes_ , lie on the unit circle in $\mathbb{C}$. At first
glance, one does not expect the complex roots to be directly observable as
they correspond to the imaginary values of physical parameters like the
magnetic field. Rather the physical consequence lies in the thermodynamic
limit, where the set of roots approach the real axis and only then would it be
of physical relevance, which is signified by phase transition.

Nevertheless, it does not mean that the complex zeros of the partition
function do not have anything to say about the system. In fact, determination
of complex Lee–Yang zeros can play a key role in studying the thermodynamics
of many body systems since the algebraic variety corresponding to these zeros
gives us information about the partition function which characterise the
thermodynamic properties of the system. By proposing a clever arrangement of a
quantum system representing an Ising model with a small number of spins, Wei
and Liu showed [3] that the complex phase in the partition function can be
represented by time evolution of a probe interacting symmetrically with the
Ising system. Using this method, the experimental observation of the Lee–Yang
zeroes on a unit circle was directly achieved in [4]. This opened up the
possibility of using a single probe to determine the full algebraic variety
containing the Lee–Yang zeros of inaccessible systems and thereby learning
about their thermodynamics. However, there are two major obstacles to this.
First of all, the phenomenon of the Lee–Yang zeros of an Ising Chain being
situated on a unit circle is a consequence of a special assumption that local
imaginary magnetic field is uniform across the lattice. Therefore it presumes
a prior knowledge about the underlying symmetry of the system. But, a general
physical system, whose internal parameters are not known to us, need not be
having necessary symmetry to satisfy this assumption. Secondly, to get
sufficient information about the full algebraic variety, we should be able to
initiate the system in a wide range of values of its internal parameters
without any prior knowledge or control over it.

In this work, we report a method by which just using a single probe and
knowing how that probe couples to the inaccessible system, one can effectively
initiate the system in any chosen value of its internal parameters and thus
determine the algebraic variety corresponding to its Lee–Yang zeros.
Considering the probe and the target is in thermal equilibrium, our protocol
works with operations on probe alone assuming no control over the system.
Finally, we experimentally demonstrate this protocol on a system of three
spin-$1/2$ nuclei using nuclear magnetic resonance (NMR) methods.

To briefly see how this works, suppose the Ising lattice has sites indexed by
$j$. The variable being complexified is $z_{j}=\mathrm{e}^{-2\beta h_{j}}$,
where $\beta$ is the inverse temperature of the system and $h_{j}$ is the
strength of the local magnetic field at site $j$. If there are a total of $n$
sites in the lattice, it was shown by Lee and Yang that the partition function
can be written in the form [5]

$\displaystyle Z\propto f(z_{1},\ldots,z_{n}),$

where $f(z_{1},\ldots,z_{n})$ is a polynomial of degree at most $1$ in each of
its variables $z_{j}$. This is called the _Lee–Yang polynomial_. The Lee–Yang
theorem actually holds for a more general statement where the roots correspond
to555See, e.g., [5] for a pedagogical coverage of the proof.

$\displaystyle|z_{1}|=|z_{2}|=\ldots=|z_{n}|=1.$ (1.1)

In the case of uniform magnetic field, all the variables
$z_{j}=\mathrm{e}^{-2\beta h_{j}}$ are equal and thus the Lee–Yang polynomial
reduces to a univariate polynomial $f(z,z,\ldots,z)=P(z)$. Hence $P(z)$ is now
defined on a single complex plane $\mathbb{C}$ and it vanishes on its unit
circle $S^{1}\subset\mathbb{C}$. It was precisely this circle that was
experimentally observed by Ref. [4].

However, this assumption of uniform magnetic field is very special and will
not be valid if we want to study a general unknown system. If we do not assume
the condition of the magnetic field being uniform, the variables $z_{j}$
remain distinct. Assuming these variables are non-zero, the values
corresponding to the vanishing of the Lee–Yang polynomial defines a set

$\displaystyle
V_{f}=\\{(z_{1},\ldots,z_{n})\in(\mathbb{C}^{*})^{n}\;|\;f(z_{1},\ldots,z_{n})=0\\},$
(1.2)

where $\mathbb{C}^{*}=\mathbb{C}\setminus\\{0\\}$. This is an _algebraic
variety_ in $(\mathbb{C}^{*})^{n}$. The set $V_{f}\subset(\mathbb{C}^{*})^{n}$
is a richer and more complicated object than the aforementioned unit circle
$S^{1}\subset\mathbb{C}$. In particular, given an algebraic variety $V$, one
can obtain its _amoeba_ [6, 7] and _co-amoeba_ 666In string theory literature,
these are known as _alga_ [8, 9]. [8, 10, 11, 12]. The relevant concepts of
amoebas and co-ameobas will be reviewed in Sec. 2. Here, we shall mention that
in fact, an alternative proof of the Lee–Yang theorem was provided using
amoebas by Passare and Tsikh in [13].

Here we present a method by which one can experimentally sample the points of
a _co-amoeba_ corresponding to any chosen point on the _amoeba_. This requires
effective preparations of the system in a wide range of its internal
parameters. Thus, by sampling the _co-amoeba_ across the _amoeba_ , one
obtains enough information about the algebraic variety attached to the
Lee–Yang polynomial of that system. Our method also allows effective
initiation of the inaccessible system in different physical situations (e.g.
ferromagnetic and anti-ferromagnetic state) and study the algebraic variety.
All these can be done just by operating and reading a single probe without
assuming much knowledge or any control over the system. As we demonstrate via
NMR architecture, it takes less than a second for the experiment to run and
sample the _co-amoeba_ corresponding to each point on _amoeba_.

The remainder of this paper is organised as follows. In Sec. 2 we review basic
notions of amoebas and co-amoebas. In Sec. 3.1 we introduce the physical
system we want to study and then in Sec 3.2 we explain the methodology of
experimentally determining the corresponding _co-amoeba_. We discuss the
experiments and how to initiate the system at the chosen value of its
parameters without any control over it corresponding to purely imaginary
fields in Sec. 4.1 and complex fields in Sec. 4.2 before concluding in Sec. 5.

##  2 Amoebas and co-amoebas

Let $f(z_{1},\ldots,z_{n})$ be a polynomial of $n$ non-zero complex variables.
It is often of interest to study the zero set777In the single-variable case
$n=1$, these are simply called the _roots_ of $f(z)$. $V_{f}$ as defined in
Eq. (1.2) is an algebraic variety. Various properties of $V_{f}$ can be
understood through its amoeba and co-amoeba which we shall now review.

The notion of an _amoeba_ was first introduced by Gelfand, Kapranov, and
Zelevinsky in their book [6]. Subsequently, the notion of _co-amoebas_ was
introduced during a talk in 2005 by Passare and Tsikh. Amoebas and coamoebas
are important objects of study because they form a link between complex
algebraic geometry and (complex) tropical geometry.

Start with a $n$-variable complex polynomial $f(z_{1},\ldots,z_{n})$. As
described in Eq. (1.2), the zero set of $f$ is the algebraic variety $V_{f}$.
To this we can associate its corresponding amoeba. First, we define the map

$\displaystyle\mathrm{Log}:(\mathbb{C}^{*})^{n}$
$\displaystyle\rightarrow\mathbb{R}^{n},$ $\displaystyle(z_{1},\ldots,z_{n})$
$\displaystyle\mapsto\left(\log|z_{1}|,\ldots,\log|z_{n}|\right).$ (2.1)

Then, the amoeba is the image of $V_{f}$ under this map. That is,
$\mathcal{A}_{f}=\mathrm{Log}\left(V_{f}\right)$. From Eq. (2.1), we see that
$\mathcal{A}_{f}=\mathrm{Log}(V_{f})$ carries information about the absolute
values of the points on the variety $V_{f}$.

(a) Amoeba.

(b) Co-amoeba.

Figure 1: The amoeba (a) and co-amoeba (b) of the polynomial
$f(z_{1},z_{2})=1+\frac{1}{2}z_{1}+\frac{1}{2}z_{2}+z_{1}z_{2}$. The amoeba in
Fig. 1(a) is generated with the aid of T. Sadykov and T. Zhukov’s website
http://amoebas.ru/index.html.

On the other hand, information about the phases of points on $V_{f}$ is
carried by the _co-amoeba_ , which is the object of interest in this paper. We
define the map

$\displaystyle\mathrm{Arg}:(\mathbb{C}^{*})^{n}$
$\displaystyle\rightarrow(S^{1})^{n},$ $\displaystyle(z_{1},\ldots,z_{n})$
$\displaystyle\mapsto\left(\mathrm{e}^{\mathrm{i}\arg
z_{1}},\ldots,\mathrm{e}^{\mathrm{i}\arg z_{n}}\right).$ (2.2)

The co-amoeba is the image of the map

$\displaystyle\mathrm{co}\mathcal{A}_{f}=\mathrm{Arg}(V_{f}).$ (2.3)

As an example, the amoeba and co-amoeba of the Lee–Yang polynomial
$f(z_{1},z_{2})=1+\frac{1}{2}z_{1}+\frac{1}{2}z_{2}+z_{1}z_{2}$ is shown in
Fig. 1.

One can identify points on the amoeba with its associated point(s) on the co-
amoeba. For instance, the subset $\\{(z_{1},z_{2})\in
V_{f}\;|\;|z_{1}|=|z_{2}|=1\\}$ is mapped under $\mathrm{Log}$ to the origin
in $\mathbb{R}^{2}$. This subset, mapped under $\mathrm{Arg}$ goes to the
boundary of the co-amoeba. On the other hand, each on $V_{f}$ for which
$|z_{1}|\neq 1$ and $|z_{2}|\neq 1$ is mapped under $\mathrm{Arg}$ to pairs of
points in the interior of the co-amoeba. We exploit this to determine the
algebraic variety by sampling the co-amoeba corresponding to different points
on the amoeba.

Figure 2: The points on the co-amoeba of $f=1+\tau z_{1}+\tau
z_{2}+z_{1}z_{2}=0$ for $\tau=0.368$. The blue curve corresponds to the
arguments of $(z_{1},z_{2})$ where $|z_{1}|=|z_{2}|=1$. The two points ‘$+$’
correspond to $|z_{1}|=0.223$ and $|z_{2}|=3.490$.

As an example to illustrate this point, we consider $\tau=0.368$ for the
polynomial $f(z_{1},z_{2})=1+\tau z_{1}+\tau z_{2}+z_{1}z_{2}$. The points
$(z_{1},z_{2})\in V_{f}$ for which $|z_{1}|=|z_{2}|=1$ are mapped under
$\mathrm{Arg}$ to the blue curves in Fig. 2. On the other hand, the points on
$V_{f}$ where $|z_{1}|=0.223$ and $|z_{2}|=3.490$ are mapped to two points
under $\mathrm{Arg}$, marked ‘$+$’ in Fig. 2.

##  3 The Ising partition function and co-amoebas

###  3.1 The two-spin Ising model

As the inaccessible system, consider an Ising model consisting of only two
spins, $A$ and $B$. Let $J$ be the coupling between the spins, and $h_{A}$ and
$h_{B}$ be the magnetic field strengths at sites $A$ and $B$, respectively.
Here $J$, $h_{A}$, and $h_{B}$ are the internal parameters of the system which
we wish to set at our will by controlling the probe alone. The Hamiltonian is

$\displaystyle
H_{AB}=-J\sigma_{z}^{A}\sigma_{z}^{B}-h_{A}\sigma_{z}^{A}-h_{B}\sigma_{z}^{B},$
(3.1)

where $\sigma_{i}^{A}$ and $\sigma_{i}^{B}$ are the respective spin’s Pauli
$i$-matrix for $i=\\{x,y,z\\}$. We denote the eigenstates of $H_{AB}$ by
$\\{\lvert 00\rangle,\lvert 01\rangle,\lvert 10\rangle,\lvert 11\rangle\\}$
with corresponding energies $E(00)$, $E(01)$, $E(10)$ and $E(11)$, where
$\\{\lvert 0\rangle,\lvert 1\rangle\\}$ represents the eigenbasis of Pauli-$z$
operator with eigenvalues $1$ and $-1$, respectively. We shall also denote the
Hilbert space spanned by these eigenstates by $\mathcal{H}_{AB}$ .

In a thermal bath of inverse temperature $\beta$, the partition function is

$\displaystyle Z(\beta,h_{A},h_{B})$
$\displaystyle=\mathrm{tr}\left(\mathrm{e}^{-\beta H_{AB}}\right)$
$\displaystyle=\mathrm{e}^{\beta J+\beta h_{A}+\beta
h_{B}}\left(1+\mathrm{e}^{-2\beta J-2\beta h_{A}}+\mathrm{e}^{-2\beta J-2\beta
h_{B}}+\mathrm{e}^{-2\beta h_{A}-2\beta h_{B}}\right).$ (3.2)

In the usual thermostatistical theory, phase transitions are signalled by the
non-analyticity of the free energy $F=-\frac{1}{\beta}\ln Z$. In particular,
this can happen at the zeroes of the partition function $Z$. From Eq. (3.2),
there are clearly no zeroes for any real values of $\beta$, $h_{A}$, or
$h_{B}$. However, if we extend $Z$ into the complex domain by letting

$\displaystyle\tau=\mathrm{e}^{-2\beta J},\quad z_{1}=\mathrm{e}^{-2\beta
h_{A}},\quad z_{2}=\mathrm{e}^{-2\beta h_{B}},$

where we allow $z_{1}$ and $z_{2}$ to take complex values, the partition
function is proportional to a complex polynomial

$\displaystyle Z$ $\displaystyle=\left(\tau
z_{1}z_{2}\right)^{-1/2}f(z_{1},z_{2}),$ $\displaystyle f(z_{1},z_{2})$
$\displaystyle=1+\tau z_{1}+\tau z_{2}+z_{1}z_{2},$ (3.3)

where $f(z_{1},z_{2})$ is the two-spin _Lee–Yang polynomial_. The zero set of
this polynomial is therefore an algebraic variety

$\displaystyle V_{f}$
$\displaystyle=\left\\{(z_{1},z_{2})\in\left(\mathbb{C}^{*}\right)^{2}\;|\;f(z_{1},z_{2})=1+\tau
z_{1}+\tau z_{2}+z_{1}z_{2}=0\right\\}.$ (3.4)

Note that the amoeba and co-amoeba depicted in Fig. 1 is the case
$\tau=\frac{1}{2}$ of $V_{f}$ above.

###  3.2 Correspondence between probe spin decoherence and Lee–Yang zeroes

The zeroes of $Z$ lie in the complex domain. To observe these complex zeroes,
we adopt the procedure of Wei and Liu [3] by introducing another spin-$1/2$
particle which acts as a probe. Its states $\\{\lvert 0\rangle,\lvert
1\rangle\\}$ span a two-dimensional Hilbert space $\mathcal{H}_{P}$. The total
space is now
$\mathcal{H}_{\mathrm{tot}}=\mathcal{H}_{P}\otimes\mathcal{H}_{AB}$. However,
contrary to [3], here we allow the probe to couple asymmetrically with $A$ and
$B$.

Suppose that the probe spin $\sigma^{P}_{z}$ couples to $\sigma_{z}^{A}$ and
$\sigma_{z}^{B}$ with coupling strengths $\lambda_{1}$ and $\lambda_{2}$,
respectively. Therefore the interaction Hamiltonian is888When writing terms
for operators which acts on different spaces, we use the notation
$\sigma^{P}_{z}\sigma^{A}_{z}=\sigma^{P}_{z}\otimes\sigma_{z}^{A}\otimes\mathbbm{1}$
and
$\sigma^{P}_{z}\sigma^{B}_{z}=\sigma^{P}_{z}\otimes\mathbbm{1}\otimes\sigma_{z}^{B}$
$H_{\mathrm{int}}=-\lambda_{1}\sigma^{P}_{z}\sigma_{z}^{A}-\lambda_{2}\sigma^{P}_{z}\sigma_{z}^{B}$.
Therefore the full Hamiltonian is

$\displaystyle H$ $\displaystyle=\mathbbm{1}\otimes
H_{AB}-\lambda_{1}\sigma^{P}_{z}\sigma^{A}_{z}-\lambda_{2}\sigma^{P}_{z}\sigma^{B}_{z}$
$\displaystyle=-J\sigma^{A}_{z}\sigma^{B}_{z}-\left(h_{A}+\lambda_{1}\sigma^{P}_{z}\right)\sigma^{A}_{z}-\left(h_{B}+\lambda_{2}\sigma^{P}_{z}\right)\sigma_{z}^{B}.$
(3.5)

This situation is depicted in Fig. 3a

Figure 3: (a) A two-spin Ising system in a thermal bath at temperature
$1/\beta$ coupled to a probe spin. (b) As the probe evolves under the
interaction with the system, it’s coherence is recorded by measuring
$\sigma_{x}^{P}$ and $\sigma_{y}^{P}$ with time.

The initial state is taken to be a quantum coherent probe spin coupled to the
Ising model in the thermal state. That is,

$\displaystyle\rho(0)$
$\displaystyle=\lvert\psi_{0}\rangle_{P}\langle\psi_{0}\rvert_{P}\otimes\frac{\exp\left(-\beta
H_{AB}\right)}{Z(\beta,h_{A},h_{B})},$ (3.6)

where $\lvert\psi_{0}\rangle$ is any coherent pure state. Of course, at the
beginning, we assume the full probe-system arrangement is in some thermal
equilibrium and from that by doing operations on probe alone we can initiate
our system in the state of Eq. (3.6) at a chosen value of
$\\{h_{A},h_{B},\beta\\}$. The details of this initiation will be discussed in
Sec. 4.1 and 4.2. Here, let us consider that the initiation in achieved and
then state evolves under the Hamiltonian mentioned in Eq. (3.5) as
$\rho(t)=\mathcal{U}(t)\rho(0)\mathcal{U}^{\dagger}(t)$, where
$\mathcal{U}(t)=\mathrm{e}^{-\frac{\mathrm{i}t}{\hbar}H}$. As we only observe
the evolution of the probe, we trace out $AB$ to get

$\displaystyle\rho_{P}(t)$ $\displaystyle=\mathrm{tr}_{AB}\rho(t)$
$\displaystyle=\frac{1}{Z(\beta,h_{A},h_{B})}\Big{[}\mathrm{e}^{-\beta
E(00)}\mathrm{e}^{-\frac{\mathrm{i}t}{\hbar}(-\lambda_{A}-\lambda_{B})\sigma_{z}^{P}}\lvert\psi_{0}\rangle_{P}\langle\psi_{0}\rvert_{P}\mathrm{e}^{\frac{\mathrm{i}t}{\hbar}(-\lambda_{A}-\lambda_{B})\sigma_{z}^{P}}$
$\displaystyle\hskip 28.45274pt+\mathrm{e}^{-\beta
E(01)}\mathrm{e}^{-\frac{\mathrm{i}t}{\hbar}(-\lambda_{A}+\lambda_{B})\sigma_{z}^{P}}\lvert\psi_{0}\rangle_{P}\langle\psi_{0}\rvert_{P}\mathrm{e}^{\frac{\mathrm{i}t}{\hbar}(-\lambda_{A}+\lambda_{B})\sigma_{z}^{P}}$
$\displaystyle\hskip 28.45274pt+\mathrm{e}^{-\beta
E(10)}\mathrm{e}^{-\frac{\mathrm{i}t}{\hbar}(\lambda_{A}-\lambda_{B})\sigma_{z}^{P}}\lvert\psi_{0}\rangle_{P}\langle\psi_{0}\rvert_{P}\mathrm{e}^{\frac{\mathrm{i}t}{\hbar}(\lambda_{A}-\lambda_{B})\sigma_{z}^{P}}$
$\displaystyle\hskip 28.45274pt+\mathrm{e}^{-\beta
E(11)}\mathrm{e}^{-\frac{\mathrm{i}t}{\hbar}(\lambda_{A}+\lambda_{B})\sigma_{z}^{P}}\lvert\psi_{0}\rangle_{P}\langle\psi_{0}\rvert_{P}\mathrm{e}^{\frac{\mathrm{i}t}{\hbar}(\lambda_{A}+\lambda_{B})\sigma_{z}^{P}}\Big{]},$
(3.7)

where $E(00)$, $E(01)$, $E(10)$, and $E(11)$ are the energy eigenvalues of the
Hamiltonian (3.1).

According to [3], the zeroes of $Z$ can be detected by measuring the coherence
of the probe spin. The relevant operator is
$S_{+}=\frac{\hbar}{2}\left(\sigma_{x}^{P}+\mathrm{i}\sigma_{y}^{P}\right)$,
where $\sigma_{x}^{P}$ and $\sigma_{y}^{P}$ are Pauli matrices acting on
$\mathcal{H}_{P}$. The spin coherence is represented by the quantity

$\displaystyle L(t)$
$\displaystyle=\frac{\hbar^{2}}{4}\left|\langle\sigma_{x}^{P}\rangle+\mathrm{i}\langle\sigma_{y}^{P}\rangle\right|^{2}.$
(3.8)

Using $\sigma_{x}^{P}=\lvert 0\rangle\langle 1\rvert+\lvert 1\rangle\langle
0\rvert$ and $\sigma_{y}^{P}=-\mathrm{i}\lvert 0\rangle\langle
1\rvert+\mathrm{i}\lvert 1\rangle\langle 0\rvert$, we compute the coherence of
the state $\rho_{P}(t)$ as given by Eq. (3.7). By direct computation,

$\displaystyle\langle\sigma_{x}^{P}+\mathrm{i}\sigma_{y}^{P}\rangle$
$\displaystyle=\mathrm{tr}\left(\sigma_{x}^{P}\rho_{P}(t)\right)+\mathrm{i}\mathrm{tr}\left(\sigma_{y}^{P}\rho_{P}(t)\right)$
$\displaystyle=\frac{2\langle\downarrow|\psi_{0}\rangle\langle\psi_{0}|\uparrow\rangle}{Z(\beta,h_{A},h_{B})}\Big{[}\mathrm{e}^{-\beta\left(-J-h_{A}-h_{B}+\frac{2\mathrm{i}t}{\hbar\beta}(\lambda_{1}+\lambda_{2})\right)}+\mathrm{e}^{-\beta\left(J-h_{A}+h_{B}+\frac{2\mathrm{i}t}{\hbar\beta}(\lambda_{1}-\lambda_{2})\right)}$
$\displaystyle\hskip
56.9055pt+\mathrm{e}^{-\beta\left(J+h_{A}-h_{B}+\frac{2\mathrm{i}t}{\hbar\beta}(-\lambda_{1}+\lambda_{2})\right)}+\mathrm{e}^{-\beta\left(-J+h_{A}+h_{B}+\frac{2\mathrm{i}t}{\hbar\beta}(-\lambda_{1}-\lambda_{2})\right)}\Big{]}.$
(3.9)

The expression in the square brackets is precisely equivalent to the partition
function $Z$, with $h_{A}$ and $h_{B}$ shifted by a translation along their
respective imaginary axes. Therefore

$\displaystyle L(t)$
$\displaystyle=\frac{\hbar^{2}}{2Z(\beta,h_{A},h_{B})}\left|\langle\downarrow|\psi_{0}\rangle\langle\psi_{0}|\uparrow\rangle
Z\left(\beta,h_{A}-\textstyle{\frac{2\mathrm{i}\lambda_{1}t}{\hbar\beta}},h_{B}-\textstyle{\frac{2\mathrm{i}\lambda_{2}t}{\hbar\beta}}\right)\right|^{2}.$
(3.10)

Therefore $L(t)$ vanishes whenever a zero of
$Z\left(\beta,h_{A}-\frac{2\mathrm{i}\lambda_{1}t}{\hbar\beta},h_{B}-\frac{2\mathrm{i}\lambda_{2}t}{\hbar\beta}\right)$
is reached. Introducing variables

$\displaystyle\tau=\mathrm{e}^{-2\beta J},\quad z_{1}=\mathrm{e}^{-2\beta
h_{A}+4\mathrm{i}\lambda_{1}t/\hbar},\quad z_{2}=\mathrm{e}^{-2\beta
h_{B}+4\mathrm{i}\lambda_{2}t/\hbar},$ (3.11)

we see that

$\displaystyle L(t)$
$\displaystyle=\mathcal{C}\left|f(z_{1},z_{2})\right|^{2},$ (3.12)

where
$\mathcal{C}=\frac{\hbar^{2}|\langle\downarrow|\psi_{0}\rangle\langle\psi_{0}|\uparrow\rangle|^{2}}{2|\tau
z_{1}z_{2}|Z(\beta,h_{A},h_{B})}$ is constant over time and $f(z_{1},z_{2})$
is the two-spin Lee–Yang polynomial

$\displaystyle f(z_{1},z_{2})=1+\tau z_{1}+\tau z_{2}+z_{1}z_{2}.$ (3.13)

In other words, the zero set of $f$ coincides with the vanishing of $L(t)$.
Furthermore, we have

$\displaystyle\log|z_{1}|=-2\beta h_{A},\quad\log|z_{2}|=-2\beta h_{B}.$
(3.14)

That is, points on the amoeba $\mathcal{A}_{f}$ parametrises the strength of
the magnetic field at sites $1$ and $2$. We also have

$\displaystyle\theta_{1}(t)=\arg
z_{1}=\frac{4\lambda_{1}t}{\hbar},\quad\theta_{2}(t)=\arg
z_{2}=\frac{4\lambda_{2}t}{\hbar}.$ (3.15)

This a line on a torus $\mathbb{T}=S^{1}\times S^{1}$ parametrised by time
$t$. In particular, its slope is

$\displaystyle m=\frac{\lambda_{2}}{\lambda_{1}}.$ (3.16)

If $m$ is a rational number, the line closes after a finite time, forming a
torus knot. However if $m$ is irrational, the line densely fills the torus.

(a) $m=1$.

(b) $m=\frac{1}{3}$.

Figure 4: Theoretical depiction of how of a co-amoeba can be sampled for
$h_{A}=h_{B}=0$, $\lambda_{1}=J/4$, $\lambda_{2}=mJ/4$, and $\beta J=0.5$, for
(a) $m=1$, and (b) $m=\frac{1}{3}$, plotted for $0\leq t\leq
6\pi\frac{\hbar}{J}$. The left panels show the plots of $t$ vs $L(t)$, while
the right panels depict the parametrised lines (3.15), overlaid with a sketch
of the coamoeba. The points when $L(t)=0$ correspond to the intersection of
the lines with the boundary of the coamoeba.

Since, at points where $f=0$, the arguments of the variables represent points
of the co-amoeba, Eq. (3.15) are straight lines in the co-amoeba plane, with
slope $\lambda_{2}/\lambda_{1}$, parametrised by $t$. As the time $t$ runs, a
point traverses the co-amoeba plane along a straight line at velocity
$\left(4\lambda_{1}/\hbar,4\lambda_{2}/\hbar\right)$. The spin coherence
$L(t)$ vanishes whenever $(\theta_{1}(t),\theta_{2}(t))$ coincides with the
corresponding point(s) on the co-amoeba.

As an example, we plot in Fig. 4 $L(t)$ vs $t$, along with the parametric
lines $\left(\theta_{1}(t),\theta_{2}(t)\right)$ for the case
$\lambda_{1}=J/4$, $\lambda_{2}=mJ/4$, $\beta=0.5$, and $h_{A}=h_{B}=0$. As
mentioned in Sec. 2, this corresponds to $|z_{1}|=|z_{2}|=0$ and these points
of $V_{f}$ is sent under Arg to the co-amoeba boundary. Therefore $L(t)$
vanishes whenever the line $\left(\theta_{1}(t),\theta_{2}(t)\right)$
intersects the amoeba boundary. We shall plot the cases $m=1$ and
$m=\frac{1}{3}$ explicitly. For the case $m=1$ (Fig. 4(a)), the line only
intersects the coamoeba boundary twice before it repeats after a period of
$2\pi\frac{\hbar}{J}\simeq 6.28\;\frac{\hbar}{J}\,\mathrm{sec}$,
correspondingly we see the graph of $L(t)$ reaching zero twice before
repeating periodically after the same period. For $m=\frac{1}{3}$ (Fig. 4(b)),
the line starts from the origin as segment A where it intersects the coamoeba
once, then continues as segment B, intersecting the coamoeba twice, and
finally as segment C, where it intersects the coamoeba one more time before it
repeats the segments A, B, C after a period of $6\pi\frac{\hbar}{J}\simeq
18.45\;\frac{\hbar}{J}\,\mathrm{sec}$.

Thus, an experimental procedure to sample the algebraic variety corresponding
to an unknown and inaccessible two-spin system can be done as follows. The
system coupled to a probe spin is initially assumed to be in thermal
equilibrium from which it is effectively instigated at a desired value of
$\beta J$, $h_{A}$,$h_{B}$ by doing operations on probe alone. Then $L(t)$ is
estimated using Eq. (3.8) by measuring $\langle\sigma_{x}^{P}(t)\rangle$ and
$\langle\sigma_{y}^{P}(t)\rangle$ of the probe spin as it evolves with time.
We then record the times when $L(t)$ vanishes. This gives a sequence, say,
$\\{t_{1},t_{2},t_{3},\ldots\\}$. By Eq. (3.15), this will form a collection
of points on the torus. As explained above, these mark the intersection points
of the line with the co-amoeba. As long as $m=\frac{\lambda_{B}}{\lambda_{A}}$
is not rational, (which, in an experimental situation, is most likely the
case), we can eventually sample enough points of the co-amoeba tied to a point
on amoeba defined by $h_{A}$ and $h_{B}$. Then we move to another point on the
amoeba plane by preparing the system at some other values of
$\\{h_{A},h_{B}\\}$ and sample the corresponding points of the _co-amoeba_.
Like this one can obtain the points of _co-amoeba_ corresponding to many
points of _amoeba_ to get a complete idea about the full algebraic variety.
Considering the probe-system arrangement is in thermal equilibrium at some
temperature initially, the challenge lies in initiating it to the state of Eq.
(3.6) at a chosen value of $\\{h_{A},h_{B},\beta\\}$ by doing operation on
probe alone which is discussed in the next section.

(a) (b)

Figure 5: (a) The molecule DBFM with three spin $1/2$ nuclei along with the 1H
NMR spectrum of the molecule. (b) Table containing the values of resonance
offsets (diagonal) and the scalar coupling constants (off-diagonal) in Hz. The
last two columns show the values of spin-lattice relaxation time constant
$T_{1}$ and effective transverse relaxation time-constant $T_{2}$.

##  4 Experiments

To get an idea about the full algebraic variety, we need to sample the points
of _co-amoeba_ connected to different points of the _amoeba_. According to Eq.
(3.14), the amoeba plane is defined by the values of $\\{h_{A},h_{B}\\}$.
Therefore determination of the algebraic variety requires sampling the points
of co-amoeba corresponding to any desired values of $h_{A}$ and $h_{B}$. We
demonstrate the process experimentally by obtaining the points of the _co-
amoeba_ for three different values of $\\{h_{A},h_{B}\\}$ corresponding to
three distinct points on the _amoeba_ plane.

We also demonstrate how we can vary the internal parameter of the system
($\beta J$) to study different physical situations like ferromagnetic ($\beta
J>0$) and anti-ferromagnetic ($\beta J<0$) regime. Considering the system and
probe to be initially in thermal equilibrium, all these can be done by
controlling and observing the probe alone. To demonstrate the method in NMR
setup, we use three spin-1/2 nuclei of ${}^{13}C$-Dibromofluoromethane (DBFM)
(see Fig. 5 (a)), dissolved in Acetone-d6, by realising ${}^{1}H$ as probe P
and ${}^{19}F$, ${}^{13}C$ as system spins $A,B$. Inside a Bruker $500$ MHz
NMR spectrometer, the sample is kept in a strong Zeeman field of $11.7~{}T$.
In a triply rotating frame [14] rotating with respect to each spin’s
individual Larmor frequency, the internal hamiltonian of the system is given
by

$\displaystyle\mathcal{H}=2\pi\left(J_{CH}I_{z}^{C}I_{z}^{H}+J_{CF}I_{z}^{C}I_{z}^{F}+J_{FH}I_{z}^{F}I_{z}^{H}\right),$
(4.1)

where $I_{i}=\sigma_{i}/2$ are the spin operators. Measured values of all
couplings along with the relaxation time constants are provided in Fig. 5b.

As we consider the system to be unknown and inaccessible, we need to set it at
a chosen values of $\beta J$, $h_{A}$, and $h_{B}$ without any knowledge of
$J_{CF}$ and without any operation on ${}^{13}C$ and ${}^{19}F$. Next, in Sec.
4.1 we demonstrate the experiments for obtaining the _co-amoeba_ in both
ferromagnetic and anti-ferromagnetic regime at the point on the _amoeba_ plane
where $\\{h_{A}=0,h_{B}=0\\}$. As can be seen from Eq. (3.11), these
correspond to Lee-Yang polynomial if the system experiences a purely imaginary
magnetic field. Finally in Sec. 4.2 we describe the experiments for non-zero
$h_{A}$ and $h_{B}$ which correspond to complex magnetic fields by same logic.

###  4.1 Experiments for Purely Imaginary Field

(a) (b) (c)
(d)

Figure 6: (a) NMR Pulse sequence to initiate the system in the state shown in
Eq. (4.3). System can be initiated at chosen value of $\beta J$ by putting
appropriate value of $\tau$. $G_{z}$ is representing a pulsed-field gradient
channel. (b) A torus surface $S^{1}\times S^{1}$. As time elapses, a point
moves on this torus surface which can be monitored from the FID signal of the
probe ${}^{1}H$. (c) Showing the point on a cartoon of _amoeba_ corresponding
to which the _co-amoeba_ is being sampled. (d) Experimental construction of
the _co-amoeba_ corresponding to different values of physical parameter
($\beta$$J$). The surface of the torus is topologically equivalent to a plane
modulo $2\pi$. Positive $J$ is in ferromagnetic regime whereas negative $J$ is
in antiferromagnetic regime. The system can be initiated in both regimes just
by varying the delay time $\tau$.

As shown in Eq. (3.6), in order to do the experiment we need to prepare three
spins in the state

$\displaystyle\rho_{PAB}^{\text{th}}=\lvert+\rangle\langle+\rvert\otimes\frac{\mathrm{e}^{\beta
J\sigma_{z}^{A}\sigma_{z}^{B}}}{{Z(\beta,0,0)}},$ (4.2)

We have written the state of Eq. (4.2) in $P,A,B$ order. The coherent state of
the probe is chosen to be $\lvert+\rangle=\frac{1}{\sqrt{2}}\left(\lvert
0\rangle+\lvert 1\rangle\right)$. The deviation density matrix [14, 15],
describing the trace-less part of the state of Eq. (4.2), can be written as

$\displaystyle\rho_{\Delta}=\frac{1}{Z(\beta,0,0)}\left[\cosh({\beta
J})I_{x}^{P}+\sinh({\beta
J})(2I_{z}^{A}I_{z}^{B}+4I_{x}^{P}I_{z}^{A}I_{z}^{B})\right],$ (4.3)

where $I_{i}=\sigma_{i}/2$. Note that the middle term $I_{z}^{A}I_{z}^{B}$ of
Eq. (4.3) does not evolve under probe-system coupling and hence ignored from
now on. In high temperature and high field approximation, the density matrix
describing the system and probe in thermal equilibrium reads
$\rho^{0}=\mathbbm{1}/8+\epsilon[I_{z}^{H}+(\gamma_{F}/\gamma_{H})I_{z}^{F}+(\gamma_{C}/\gamma_{H})I_{z}^{C}]$,
where $\gamma_{i}$ is the gyro-magnetic ratio of nucleus $i$ and the purity
factor $\epsilon\approx 10^{-5}$. From this state, the desired initial state
of Eq. (4.3) is achieved up to a harmless identity population back-ground by
applying the pulse program shown in Fig. 6a. We stress that by this method the
system can be initiated at any desired value of $\beta J$ by varying the delay
time $\tau$ suitably. In fact, these two quantities are related by the
equation:
$~{}\cos{(\pi(J_{PA}+J_{PB})\tau)}/\cos{(\pi(J_{PA}-J_{PB})\tau)}=\exp{(2\beta
J)}$. No gate needs to be operated on the system and no knowledge of the value
of $J_{FC}$ is required for this.

Once the desired state is prepared for a chosen value of $\beta J$, the system
is let to evolve under probe-system ($H-F$ and $H-C$) coupling. We get the
continuous time evolution of the probe’s coherence, as described in Eq.
(3.10), directly from the recorded free induction decay (FID) signal of
${}^{1}H$. As $X$ and $Y$ magnetization of ${}^{1}H$ (quantified as
$\langle\sigma_{x}^{P}\rangle$ and $\langle\sigma_{y}^{P}\rangle$ in Eq. (3.9)
respectively) evolves in time, it traces a path on the surface of a $2$-torus
(shape is depicted in Fig. 6b) as described in Sec. 3.2. It can be seen
analytically that by evolving the state of Eq. (4.3) under $J_{FH}$ and
$J_{CH}$, $\langle\sigma_{y}^{P}\rangle$ will always remain zero. Therefore
just recording the zeros of $\langle\sigma_{x}^{P}\rangle$ corresponding to
real part of FID will suffice. To avoid a closed torus knot, we need to have
the ratio $J_{FH}/J_{CH}$ to be irrational which is satisfied by our molecule
as can be seen from the table in Fig. 5b. Thereby recording the time points in
which the FID reaches zero, we collect enough Lee-Yang zeros to sample the
boundary of the co-amoeba as shown in Fig. 6c. We instigated the Ising system
at three different values of $\beta J$ just by doing probe operation and
determined corresponding _co-amoebas_. These points of the _co-amoebas_
correspond to the origin on the _amoeba_ plane (see Fig. 7d).

###  4.2 Experiments for Complex Fields

To sample the _co-amoeba_ for the points on the _amoeba_ plane other than the
origin, we need to have our initial state of Eq. (3.6) with non-zero $h_{A}$
and $h_{B}$. According to Eq. 3.11, these correspond to Lee-Yang polynomial
when the system is kept under complex magnetic field. The deviation density
matrix for this state can be written as

$\displaystyle\rho_{\Delta}\propto
c_{1}I_{x}^{P}+c_{2}2I_{x}^{P}I_{z}^{A}+c_{3}2I_{x}^{P}I_{z}^{B}+c_{4}4I_{x}^{P}I_{z}^{A}I_{z}^{B},~{}~{}~{}\text{where,}$
(4.4) $\displaystyle c_{1}=$ $\displaystyle\cosh{(\beta h_{A})}\cosh{(\beta
h_{B})}\cosh{(\beta J)}-\sinh{(\beta h_{A})}\sinh{(\beta h_{B})}\sinh{(\beta
J)},$ $\displaystyle c_{2}=$ $\displaystyle\cosh{(\beta h_{A})}\sinh{(\beta
h_{B})}\sinh{(\beta J)}-\sinh{(\beta h_{A})}\cosh{(\beta h_{B})}\cosh{(\beta
J)},$ $\displaystyle c_{3}=$ $\displaystyle\sinh{(\beta h_{A})}\cosh{(\beta
h_{B})}\sinh{(\beta J)}-\cosh{(\beta h_{A})}\sinh{(\beta h_{B})}\cosh{(\beta
J)},$ $\displaystyle c_{4}=$ $\displaystyle\sinh{(\beta h_{A})}\sinh{(\beta
h_{B})}\cosh{(\beta J)}-\cosh{(\beta h_{A})}\cosh{(\beta h_{B})}\sinh{(\beta
J)}.$

(a) (b) (c)

(d)

Figure 7: (a) NMR pulse sequence for initiating the system in state of Eq.
4.4. Parameters $h_{A}$,$h_{B}$ and $\beta J$ can be chosen by putting
appropriate values of $\theta^{1}$,$\theta^{2}$ and $\tau$. (b) Experimental
data corresponding to the case where $h_{A}$ and $h_{B}$ are in opposite
direction. Red and black points correspond to zeros of
$\langle\sigma_{x}^{P}\rangle$ (real signal) and
$\langle\sigma_{y}^{P}\rangle$ (imaginary signal) as measured experimentally.
Solid black line is the simulation which lies close to the experimental points
shown in black. Intersection of real and imaginary zeros are depicting two
Yang-Lee zeros. (c) Experimental data corresponding to the case where
$h_{A}=0.1$ and $h_{B}=0.1$ ($\beta J=0.5$) are in same direction. There is no
Yang-Lee zero in this case as real and imaginary zeros do not intersect. (d)
Qualitatively showing the two points on the _amoeba_ plane for which
corresponding _co-amoeba_ is sampled experimentally. The _amoeba_ shown here
is just a cartoon and should not be taken quantitatively.

Again, after starting from the thermal equilibrium as before, this state of
Eq. (4.4) can be prepared using the pulse sequence shown in Fig. 7a. System
can be effectively initiated at any desired value of internal parameters
$h_{A},h_{B}$ and $\beta J$ by putting appropriate values of
$\theta^{1}$,$\theta^{2}$ and $\tau$. Varying $\\{h_{A},h_{B}\\}$ allows us to
sample _co-amoeba_ connected to different points of the _amoeba_ plane while
by varying $\beta J$ on can go from ferromagnetic to anti-ferromagnetic
regime. Again, this requires no prior knowledge of $J_{CF}$ and only probe
operations. Once, the state is prepared, we let the probe to evolve under its
couplings with the Ising system and record its FID. The times when
$\langle\sigma_{x}^{P}\rangle$ and $\langle\sigma_{y}^{P}\rangle$ become zero
are recorded and its corresponding points $(\theta_{1}(t),\theta_{2}(t))$ are
marked on the co-amoeba plane. As can be seen from Eq. (3.7) and Eq. (3.8),
simultaneous zeros of real part of FID (corresponding to
$\langle\sigma_{x}^{P}\rangle$) and imaginary part of FID (corresponding to
$\langle\sigma_{y}^{P}\rangle$) lead to Lee-Yang zero. As can be seen from the
simulation of Fig. 1(a), Lee-Yang zeros occur only when the local fields
$h_{A}$ and $h_{B}$ are in different direction. Experimental results confirm
the same as shown in Fig. 7b and 7c.

We stress that for both only pure and complex magnetic fields, since the _co-
amoeba_ is constructed directly from NMR FID, each experiment takes less than
a second to complete. Thus these experiments can be repeated easily to get the
_co-amoebas_ for many different points on the _amoeba_ plane to get the full
algebraic variety.

##  5 Conclusion

The algebraic variety corresponding to Lee–Yang polynomial of a system is
defined as the set of values of the complexified physical variables for which
the partition function vanishes. Knowledge about this set sheds light on the
partition function of the system and hence characterizes its thermodynamics.
In this work, we have presented a method to experimentally determine the
algebraic variety of an inaccessible system. The method works by initiating
the system at chosen values of internal parameters and then obtaining the _co-
amoeba_ corresponding to the desired point on the _amoeba_ plane. This can be
repeated to cover many points on the _amoeba_ plane to get good idea about the
algebraic variety. The initiation is achieved through a single probe only
without any knowledge of the system parameters and without any control over
it. The method can be generalized to study much larger inaccessible systems.

## Acknowledgments

A. C. acknowledges Kunal Arora for discussions. T. S. M. acknowledges funding
from DST/ ICPS/QuST/2019/Q67 and I-HUB QTF. M. N. is supported by Xiamen
University Malaysia Research Fund (Grant No. XMUMRF/2020-C5/IMAT/0013). Y.-K.
L. is supported by Xiamen University Malaysia Research Fund (Grant no.
XMUMRF/2021-C8/IPHY/0001).

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