# A Survey of optimal polynomial approximants, applications to digital filter
design, and related open problems

Catherine Bénéteau  Department of Mathematics, University of South Florida,
4202 E. Fowler Avenue, Tampa, Florida 33620-5700, USA<EMAIL_ADDRESS>and
Raymond Centner  Department of Mathematics, University of South Florida, 4202
E. Fowler Avenue, Tampa, Florida 33620-5700, USA<EMAIL_ADDRESS>
###### Abstract.

In the last few years, the notion of optimal polynomial approximant has
appeared in the mathematics literature in connection with Hilbert spaces of
analytic functions of one or more variables. In the 70s, researchers in
engineering and applied mathematics introduced least-squares inverses in the
context of digital filters in signal processing. It turns out that in the
Hardy space $H^{2}$ these objects are identical. This paper is a survey of
known results about optimal polynomial approximants. In particular, we will
examine their connections with orthogonal polynomials and reproducing kernels
in weighted spaces and digital filter design. We will also describe what is
known about the zeros of optimal polynomial approximants, their rates of
decay, and convergence results. Throughout the paper, we state many open
questions that may be of interest.

###### 2010 Mathematics Subject Classification:

Primary 30Hxx; Secondary 30Jxx. Key words: optimal polynomial approximants,
digital filters, Dirichlet spaces, reproducing kernels, orthogonal polynomials

##  1\. Introduction

Consider a collection of Hilbert spaces of analytic functions of the unit disk
$\mathbb{D}$ defined as follows. Given weights
$\\{\omega_{k}\\}_{k\in\mathbb{N}}$ satisfying $\omega_{0}=1$, $\omega_{k}>0$
and $\omega_{k}/\omega_{k+1}\rightarrow 1$ as $k\rightarrow\infty$, we define
the weighted Hardy space $H^{2}_{\omega}$ as the space of functions
$f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ analytic in $\mathbb{D}$ such that

$\|f\|^{2}_{\omega}=\sum_{k=0}^{\infty}|a_{k}|^{2}\omega_{k}<\infty.$

$H^{2}_{\omega}$ is a Hilbert space with inner product

$\langle
f,g\rangle_{\omega}=\sum_{k=0}^{\infty}a_{k}\overline{b_{k}}\omega_{k},$

for $g(z)=\sum_{k=0}^{\infty}b_{k}z^{k}.$ If $\omega_{k}=(k+1)^{\alpha}$ for a
real parameter $\alpha,$ the spaces $H^{2}_{\omega}={\mathcal{D}}_{\alpha}$
are called Dirichlet-type spaces and include the Hardy space
$H^{2}=\mathcal{D}_{0},$ the classical Dirichlet space
$\mathcal{D}_{1}=\mathcal{D},$ and the Bergman space $\mathcal{D}_{-1}=A^{2}.$
In the case of the Hardy space $H^{2}$, we omit the subscript $\omega$ and
write $\langle\cdot,\cdot\rangle$ for the inner product. Given a function
$f\in H^{2}_{\omega}$ not identically $0$ and $n\in\mathbb{N},$ we define the
$n$-th optimal polynomial approximant (opa) of $1/f$ in $H^{2}_{\omega}$ to be
the polynomial $q_{n}$ that minimizes $\|pf-1\|_{\omega}$ among all
polynomials $p$ of degree at most $n.$ Notice that if we denote by
$\mathcal{P}_{n}$ to be the set of all polynomials of degree at most $n$, then
$f\cdot\mathcal{P}_{n}$ is a finite dimensional subspace of $H^{2}_{\omega}$,
and $q_{n}f$ is simply the projection of $1$ onto that subspace. Therefore,
$q_{n}$ always exists and is unique.

These polynomials were defined in [1] in the spaces $\mathcal{D}_{\alpha}$ in
connection with cyclicity: a function $f\in H^{2}_{\omega}$ is called cyclic
if its polynomial multiples are dense in the space. In particular, if $f$ is
cyclic, there must be a sequence of polynomial multiples of $f$ that
approximate $1$, and thus, the opas $q_{n}$ are the ones that achieve this
approximation at the fastest rate in terms of the degree. On the other hand,
if $1$ can be approximated by polynomial multiples of $f$, then so can any
polynomial, and since polynomials are dense in $H^{2}_{\omega}$, $f$ must be
cyclic. Therefore $f$ is cyclic if and only if the opas $q_{n}$ satisfy
$\|q_{n}f-1\|_{\omega}\rightarrow 0$ as $n\rightarrow\infty.$ Thus, one can
view cyclicity through the lens of the behavior of the optimal polynomial
approximants.

The authors of [1] were hoping to be able to shed some light on the Brown and
Shields conjecture (see [9]) using opas. The Brown and Shields conjecture
states that for a function $f$ in the classical Dirichlet space $\mathcal{D},$
if $f$ is outer and has boundary zero set $Z(f)$ with logarithmic capacity
$0$, then $f$ is cyclic. (The converse is known to be true.) Although this
initial hope has not yet proved successful, many other interesting problems
related to optimal polynomial approximants arose, in particular in connection
with inner functions (see, e.g., [3, 15, 24]).

It turns out that optimal polynomial approximants had been introduced much
earlier in the engineering literature in the 70s, in connection with signal
processing and digital filter construction. They were first introduced by
Robinson ([23]) and referred to as least-squares inverses. Robinson wanted to
find the inverse under discrete convolution of a finite-length minimum-delay
wavelet that best approximates the unit spike. From a mathematical point of
view, this meant, given a finite sequence $b:=(b_{0},b_{1},\ldots,b_{n})$ of
real numbers, where the largest coefficient in magnitude is $b_{0}$ (the so-
called “minimum delay”), find a “wavelet”, i.e., a sequence
$a:=(a_{0},a_{1},\ldots,a_{m})$ of real numbers such that the difference
between the convolution $a*b$ and the unit spike $(1,0,\ldots,0)$ has smallest
$\ell^{2}$-norm. This is equivalent to the problem of finding the optimal
polynomial approximant of $1/f$ for a polynomial $f$ with $f(0)\neq 0$. In
[12], Chui examined the _double least-squares inverse_ of a polynomial $f$ of
degree $n$ by first finding the $k$-th opa $q_{k}$ of $1/f$ and then finding
the $n$-th opa $p_{n,k}$ of $1/q_{k}$. He studied the zeros of the double
least-squares inverses, noticed the connection with orthogonal polynomials
(which had also been discussed for polynomials with real coefficients in
[17]), and proved that $p_{n,k}$ converges to $f$ in the finite-dimensional
space of polynomials of degree $n$ if and only if $f$ has no zeros in the open
unit disk. He went on to discuss some generalizations to the case when $f\in
H^{2},$ which was further studied by Izumino in [20]. In several variables,
least-squares inverses were studied by a number of authors in the 70s in
connection with recursive digital filters (see, e.g., [17, 27]). Two recent
papers of Sargent and Sola ([25, 26]) discuss some of these developments and
examine the more complicated relationship between optimal polynomial
approximants and orthogonal polynomials in several variables.

This paper is a survey of some of the known results about optimal polynomial
approximants, applications to digital filter theory, and related open
problems. One of the goals of the paper is, as also discussed in [25], to
connect the two sets of literature on these topics. In Section 2, we describe
how to compute opas, give some basic examples, and explain how opas are
connected to orthogonal polynomials and reproducing kernels for weighted
spaces. In Section 3, we discuss how opas were used in the design of digital
filters in the engineering literature of the 70s. In Section 4, we survey some
known results about the zeros of opas in the single and multivariable case and
state several open questions. Finally, in Section 5, we discuss rates of decay
and pointwise convergence results.

##  2\. Existence of opas and connections with orthogonal polynomials and
reproducing kernels

Let us now turn to a discussion of how to compute opas and their connection
with classical objects in analysis. We have already seen that for any function
$f\in H^{2}_{\omega}\setminus\\{0\\},$ and any $n\in\mathbb{N},$ the $n$-th
opa $q_{n}$ always exists. It is also not hard to see that the coefficients of
$q_{n}$ can be computed by a set of linear equations ([1, 16]), as stated in
the following theorem.

######  Theorem 2.1.

For any function $f\in H^{2}_{\omega}\setminus\\{0\\}$, let
$q_{n}(z)=\sum_{j=0}^{n}a_{j}z^{j}$ denote the $n$-th opa of $1/f$. Define the
vectors $a=\begin{bmatrix}a_{0}&a_{1}&\dots&a_{n}\end{bmatrix}^{T}$ and
$y=\begin{bmatrix}\overline{f(0)}&0&\dots&0\end{bmatrix}^{T}$. If $B$ is the
$(n+1)\times(n+1)$ matrix given by $B_{jk}=\langle
z^{k}f,z^{j}f\rangle_{\omega}$, where $0\leq j,k\leq n$, then

$Ba=y.$

###### Proof.

Note that by definition, $q_{n}$ is the projection of $1$ onto
$\mathcal{P}_{n}\cdot f$. By orthogonality in $H^{2}_{\omega}$, we thus have
that

$\displaystyle 0$ $\displaystyle=\langle q_{n}f-1,z^{j}f\rangle_{\omega}$
$\displaystyle=\langle\sum_{k=0}^{n}a_{k}z^{k}f-1,z^{j}f\rangle_{\omega}$
$\displaystyle=\sum_{k=0}^{n}a_{k}\langle
z^{k}f,z^{j}f\rangle_{\omega}-\langle 1,z^{j}f\rangle_{\omega}$
$\displaystyle=\sum_{k=0}^{n}a_{k}\langle
z^{k}f,z^{j}f\rangle_{\omega}-\delta_{j,0}\overline{f(0)}$

for $j=0,\cdots,n$. This is equivalent to the assertion $Ba=y$.

∎

Let’s have a look at an example for $H^{2}$ in which the form of the matrix is
simple enough to allow us to determine an explicit formula for the
coefficients.

######  Example 2.2.

Let $q_{n}$ denote the $n$-th opa of $1/f$, where $f(z)=1-z$. Then

$q_{n}(z)=\sum_{j=0}^{n}\frac{n+1-j}{n+2}z^{j}.$

###### Proof.

Let $q_{n}(z)=\sum_{j=0}^{n}a_{j}z^{j}$ and let $a$ and $y$ denote the vectors
$a=\begin{bmatrix}a_{0}&a_{1}&a_{2}&\dots&a_{n}\end{bmatrix}^{T}$ and
$y=\begin{bmatrix}1&0&0&\dots&0\end{bmatrix}^{T}$. From Theorem 2.1, we have

$Ba=y,$

where $B$ is the $(n+1)\times(n+1)$ matrix with

$\displaystyle B_{j,k}$ $\displaystyle=\langle z^{k}f,z^{j}f\rangle$
$\displaystyle=\langle z^{k}(1-z),z^{j}(1-z)\rangle$ $\displaystyle=\langle
z^{k}-z^{k+1},z^{j}-z^{j+1}\rangle$ $\displaystyle=\langle
z^{k},z^{j}\rangle+\langle z^{k+1},z^{j+1}\rangle-\langle
z^{k+1},z^{j}\rangle-\langle z^{k},z^{j+1}\rangle$

for $0\leq j,k\leq n$. Therefore, we see that

$B_{j,k}=\begin{cases}2&\text{if $j=k$}\\\ -1&\text{if $|j-k|=1$}\\\
0&\text{otherwise}\end{cases}$

and note that $a_{0}=1/2$ whenever $n=0$. If $n=1$, we get

$\begin{cases}2a_{0}-a_{1}=1\\\ -a_{0}+2a_{1}=0.\end{cases}$

For $n\geq 2$, the coefficients $\\{a_{j}\\}_{j=0}^{n}$ satisfy

$\begin{cases}2a_{0}-a_{1}=1\\\ -a_{j}+2a_{j+1}-a_{j+2}=0\qquad 0\leq j\leq
n-2\\\ -a_{n-1}+2a_{n}=0.\end{cases}$

These conditions can be summarized by the matrix equation

$\begin{bmatrix}2&-1&0&0&\dots&0\\\ -1&2&-1&0&\dots&0\\\ 0&-1&2&-1&\dots&0\\\
\vdots&&\ddots&\ddots&\ddots&\vdots\\\ 0&\dots&0&-1&2&-1\\\
0&\dots&0&0&-1&2\end{bmatrix}\begin{bmatrix}a_{0}\\\ a_{1}\\\ a_{2}\\\
a_{3}\\\ \vdots\\\ a_{n}\end{bmatrix}=\begin{bmatrix}1\\\ 0\\\ 0\\\ 0\\\
\vdots\\\ 0\end{bmatrix}.$

It is straightforward to check that $\\{\frac{n+1-j}{n+2}\\}_{j=0}^{n}$
satisfies these conditions.

∎

This computation was generalized in [16], where the authors show that the opa
$q_{n}$ of $1/(1-z)$ in $H^{2}_{\omega}$ is equal to

$q_{n}(z)=\sum_{k=0}^{n}\left(1-\frac{\sum_{j=0}^{k}\frac{1}{\omega_{j}}}{\sum_{j=0}^{n+1}\frac{1}{\omega_{j}}}\right).$

In $H^{2},$ formulas for the opa of $1/(1-z)^{a}$ where $Re(a)>0$ are obtained
in [8], but in general, it is very difficult to calculate opas explicitly.
Computations of opas focused on the case when $f$ is a polynomial are
considered in [7]. From the point of view of classical analysis, it is easy to
see that one way to calculate $q_{n}f$ is to consider the set of vectors
$\\{f,zf,z^{2}f,\ldots,z^{n}f\\},$ use the Gram-Schmidt process to find an
orthonormal basis of the finite dimensional space $f\cdot\mathcal{P}_{n},$ and
express $q_{n}f$ in that basis. This gives rise to orthonormal polynomials and
is the content of the following theorem, discussed in [5].

######  Theorem 2.3 ([5]).

Let $f\in H^{2}_{\omega}\setminus\\{0\\}$, $n\in\mathbb{N},$ and let $q_{n}$
be the $n$-th opa of $1/f.$ For each $k=0,1,\ldots n,$ let $\varphi_{k}$ be a
polynomial of degree $k$ such that $\\{\varphi_{k}f\\}_{k=0}^{n}$ is an
orthonormal basis of $f\cdot\mathcal{P}_{n}.$ Then

$q_{n}(z)=\overline{f(0)}\sum_{k=0}^{n}\overline{\varphi_{k}(0)}\varphi_{k}(z).$
(1)

###### Proof.

Since $q_{n}f\in f\cdot\mathcal{P}_{n}$ and $\\{\varphi_{k}f\\}_{k=0}^{n}$ is
an orthonormal basis of $f\cdot\mathcal{P}_{n},$ we can express $q_{n}f$ in
that basis:

$q_{n}(z)f(z)=\sum_{k=0}^{n}\langle
q_{n}f,\varphi_{k}f\rangle_{\omega}\varphi_{k}(z)f(z).$

But $q_{n}f$ is the projection of $1$ onto $f\cdot\mathcal{P}_{n},$ and
therefore $\langle q_{n}f,\varphi_{k}f\rangle_{\omega}=\langle
1,\varphi_{k}f\rangle_{\omega}.$ By the defintion of the inner product in
$H^{2}_{\omega}$, this gives

$q_{n}(z)f(z)=\sum_{k=0}^{n}\overline{f(0)\varphi_{k}(0)}\varphi_{k}(z)f(z),$

and therefore, (1) follows. ∎

The polynomials $\varphi_{k}$ can be thought of as orthonormal polynomials in
the Hilbert space weighted by $f$, in the sense that
$\langle\varphi_{k}f,\varphi_{j}f\rangle_{\omega}=\delta_{kj}.$ In particular,
if we restrict to the classical space $H^{2}$, this is the same as saying that
the polynomials $\varphi_{k}$ are orthonormal polynomials in the weighted
space $H^{2}(\mu)$ where the measure $d\mu=\frac{1}{2\pi}|f|^{2}d\theta.$ This
approach was discussed in [13] for the Hardy space and in [5] for Dirichlet-
type spaces.

Notice also that since $\\{\varphi_{k}f\\}_{k=0}^{n}$ is an orthonormal basis
of $f\cdot\mathcal{P}_{n}$,
$\sum_{k=0}^{n}\overline{f(0)\varphi_{k}(0)}\varphi_{k}(z)f(z)=:K_{n}(z,0)$ is
the reproducing kernel for $f\cdot\mathcal{P}_{n}$ at $0:$ this means that for
any polynomial $q\in\mathcal{P}_{n}$, $\langle
qf,K_{n}(\cdot,0)\rangle_{\omega}=q(0)f(0).$ This can also be seen directly,
since

$\langle qf,q_{n}f\rangle_{\omega}=\langle qf,1\rangle_{\omega}=q(0)f(0).$

Thus $q_{n}(z)f(z)=K_{n}(z,0).$

In the particular case of $H^{2},$ the $n$-th opa $q_{n}$ is even more tightly
connected to the $n$-th orthonormal polynomial, as is well-known (see, e.g.,
[18, 28]). The authors of [5] used this connection to express the opas in
terms of weighted orthonormal polynomials.

######  Theorem 2.4 ([5]).

Let $f\in H^{2}\setminus\\{0\\},$ $n\in\mathbb{N},$ and let $q_{n}$ be the
$n$-th opa of $1/f.$ For each $k=0,1,\ldots n,$ let $\varphi_{k}$ be a
polynomial of degree $k$ such that $\\{\varphi_{k}f\\}_{k=0}^{n}$ is an
orthonormal basis of $f\cdot\mathcal{P}_{n}.$ Let
$\varphi^{*}_{n}(z)=z^{n}\overline{\varphi_{n}(1/\bar{z})}.$ Then
$q_{n}(z)=\overline{f(0)}\hat{\varphi}_{n}(n)\varphi_{n}^{*}(z).$

###### Proof.

Let $q_{n}(z)=\sum_{k=0}^{n}a_{k}z^{k}$ and consider the weighted space
$H^{2}(\mu)$, where $d\mu=\frac{1}{2\pi}|f|^{2}d\theta$. In light of Theorem
2.1, we see that

$\displaystyle\delta_{j,0}f(0)$ $\displaystyle=\sum_{k=0}^{n}\langle
z^{j}f,z^{k}f\rangle\overline{a_{k}}$ $\displaystyle=\langle
z^{j}f,\sum_{k=0}^{n}a_{k}z^{k}f\rangle$ $\displaystyle=\langle
z^{j},q_{n}\rangle_{\mu}$ $\displaystyle=\langle
z^{n}\overline{q_{n}},z^{n-j}\rangle_{\mu}$ $\displaystyle=\langle
q_{n}^{*},z^{n-j}\rangle_{\mu}.$

Now since $\\{\varphi_{k}\\}_{k=0}^{n}$ is an orthonormal basis of
$H^{2}(\mu)$, it follows that

$\displaystyle q_{n}^{*}(z)$ $\displaystyle=\sum_{k=0}^{n}\langle
q_{n}^{*},\varphi_{k}\rangle_{\mu}\varphi_{k}(z)$ $\displaystyle=\langle
q_{n}^{*},\varphi_{n}\rangle_{\mu}\varphi_{n}(z)$
$\displaystyle=f(0)\overline{\hat{\varphi}_{n}(n)}\varphi_{n}(z).$

Therefore,

$\displaystyle q_{n}(z)$
$\displaystyle=z^{n}\overline{q_{n}^{*}(1/\overline{z})}$
$\displaystyle=\overline{f(0)}\hat{\varphi}_{n}(n)z^{n}\overline{\varphi_{n}(1/\overline{z})}$
$\displaystyle=\overline{f(0)}\hat{\varphi}_{n}(n)\varphi_{n}^{*}(z).$

∎

It is known from the theory of orthogonal polynomials that the orthonormal
polynomials $\varphi_{n}$ have all their zeros inside the open unit disk
$\mathbb{D},$ and therefore the opas $q_{n}$ have no zeros in the closed disk.
In particular, this connection was important for designing what are called
stable filters, which we discuss in the next section.

##  3\. Digital Filter Design

###  3.1. Introduction to filters

Several problems in engineering ultimately depend on a system’s response to an
input. In the case of a digital system, the input is given as a sampling
sequence $\\{x(n)\\}_{n=-\infty}^{\infty}$, and the output
$\\{y(n)\\}_{n=-\infty}^{\infty}$ can often be described by a difference
equation

$y(n)=\sum_{k=0}^{M}b_{k}x(n-k)-\sum_{j=1}^{N}a_{j}y(n-j),$ (2)

where the coefficients $a_{j}$ and $b_{k}$ are real numbers.

If the coefficients remain constant over time, the system is known as linear
time-invariant, or LTI. If the $a_{j}$’s are not all zero, then the system is
referred to as recursive. This means that one or more of the system’s output
is used as an input. Now, if we consider an input sequence
$\\{x(n)\\}_{n=-\infty}^{\infty}$ that is bounded, it seems problematic in
practice for $|y(n)|$ to increase without bound as $n\rightarrow\infty$.
Therefore, it is of interest to seek for properties of a system that preserve
boundedness. A system in which a bounded input yields a bounded output is
called  BIBO stable.

In order to facilitate our discussion of filters, we will assume that our
input sequences $\\{x(n)\\}_{n=-\infty}^{\infty}$ have the property that
$x(n)=0$ for $n<0$. A sequence with this property is known as causal.
Moreover, we will assume that the input sequences are exponentially bounded.
That is, we assume that

$|x(n)|\leq K^{n},\enskip n\geq n_{0}$

for some constant $K$ and some integer $n_{0}$. Now, to better understand the
relationship between the input and output sequences, we make use of the
following operator. For any causal sequence $\\{a(n)\\}_{n=-\infty}^{\infty}$
that’s exponentially bounded, consider the mapping

$\\{a(n)\\}_{n=-\infty}^{\infty}\mapsto\sum_{n=0}^{\infty}a(n)z^{-n}.$

This mapping is known as the z-transform. It is a linear operator from the
space of exponentially bounded causal sequences onto the space of functions
analytic at $\infty$. The $z$-transform of a sequence
$\\{a(n)\\}_{n=-\infty}^{\infty}$ has a region of convergence (ROC) given by
$z\in\hat{\mathbb{C}}$ such that

$\limsup\sqrt[n]{|a(n)|}<|z|.$ (3)

With this, the sum and product of two transformed sequences are defined to be
in the intersection of both ROCs, and the product is given by the expression

$\Bigg{(}\sum_{n=0}^{\infty}a(n)z^{-n}\Bigg{)}\Bigg{(}\sum_{n=0}^{\infty}b(n)z^{-n}\Bigg{)}=\sum_{n=0}^{\infty}\Bigg{(}\sum_{k=0}^{n}a(k)b(n-k)\Bigg{)}z^{-n}.$

The $z$-transform has many properties that make it useful in the analysis of
digital systems. In particular, if $A(z)$ is the $z$-transform of the sequence
$\\{a(n)\\}_{n=-\infty}^{\infty}$, then the $z$-transform of the sequence
$\\{c(n)=a(n-N):-\infty<n<\infty\\}$ is $z^{-N}A(z)$ for any $N\in\mathbb{N}$.
Therefore, by applying the $z$-transform to both sides of (2), we see that the
$z$-transforms of the input and output are related by the equation

$Y(z)=H(z)X(z),$

where $H(z)$ is given by the rational function

$H(z)=\frac{\sum_{k=0}^{M}b_{k}z^{-k}}{1+\sum_{j=1}^{N}a_{j}z^{-j}}.$ (4)

We use $X(z)$ and $Y(z)$ to represent the the $z$-transforms of the input and
output sequences, respectively. The rational function $H$ is known as the
transfer function, or filter, of the system. In the case of a recursive
system, the transfer function is commonly called an infinite impulse response
filter, or IIR filter.

For the purpose of our discussion, we will only be considering recursive LTI
systems. For simplicity, we refer to a BIBO stable system as stable. Likewise,
we refer to the filter of a BIBO stable system as stable. The goal of Section
3 is to demonstrate how optimal polynomial approximants are used in designing
a stable filter. It’s worth noting that although we will be considering
filters of a single variable, several applications are concerned with filters
of multiple variables. As mentioned in [19], the processing of medical
pictures, satellite photographs, radar and sonar maps, seismic data mappings,
gravity waves data, and magnetic recordings are examples in which 2D signal
processing is needed. Here, we are concerned with designing filters $H(z,w)$
in $\mathbb{C}^{2}$ of analogous stability.

###  3.2. Stability

An important part of designing a digital filter is ensuring that the filter is
stable. The stability of the filter will prevent the magnitude of the output
from increasing without bound, which could be damaging to the physical system.
Therefore, we seek for properties of the expression in (4) which guarantee
stability. If we define the polynomials $A(z)=1+\sum_{j=1}^{N}a_{j}z^{j}$ and
$B(z)=\sum_{k=0}^{M}b_{k}z^{k}$, we see that

$H(z)=\frac{z^{N-M}B^{*}(z)}{A^{*}(z)},$ (5)

where $A^{*}(z)$ and $B^{*}(z)$ are the reverse polynomials of $A(z)$ and
$B(z)$, respectively. From (5), it is easy to check that $H(z)$ is a rational
function analytic at $\infty$. Therefore, $H(z)$ is the $z$-transform of some
causal sequence $\\{h(n)\\}_{n=-\infty}^{\infty}$, and we write

$H(z)=\sum_{n=0}^{\infty}h(n)z^{-n}$ (6)

for some specified ROC.

Now if we assume that the poles of $H(z)$ are contained in the disk
$\mathbb{D}$, the expression in (6) must be vaild on $\mathbb{T}$. Since the
series defined by $\sum_{n=0}^{\infty}|h(n)|z^{-n}$ has the same ROC, it
follows that

$\sum_{n=0}^{\infty}|h(n)|<\infty.$ (7)

This observation leads us to the following theorem.

######  Theorem 3.1.

A filter $H(z)$ is stable if its poles are contained in the disk $\mathbb{D}$.

###### Proof.

Let $\\{x(n)\\}_{n=-\infty}^{\infty}$ be an input sequence with $|x(n)|\leq M$
for all $n$. If the poles of $H(z)$ are contained in $\mathbb{D}$, then (7)
holds. Consequently, the output sequence $\\{y(n)\\}_{n=-\infty}^{\infty}$ is
bounded with

$\displaystyle|y(n)|$ $\displaystyle=\Bigg{|}\sum_{k=0}^{n}h(k)x(n-k)\Bigg{|}$
$\displaystyle\leq M\sum_{n=0}^{\infty}|h(n)|$

for all $n$. By definition, $H(z)$ must be stable. ∎

This theorem gives a sufficient condition for a filter to be stable. However,
it’s important to note that a filter with a pole outside of $\mathbb{D}$ need
not be stable. As an example, consider the function

$H(z)=\frac{1}{z-1}.$

For all $|z|>1$, this function is represented by the series

$H(z)=\sum_{n=1}^{\infty}z^{-n}.$

If we consider the input defined by $x(n)=1$ for $n\geq 0$, the magnitude of
the output is given by

$|y(n)|=\Bigg{|}\sum_{k=0}^{n}h(k)x(n-k)\Bigg{|}=n,$

which clearly increases without bound as $n\rightarrow\infty$.

###  3.3. Frequency Response

Many problems are concerned with how a system responds to a sinusoidal input.
This is particularly evident in audio equalizing, where the input function
represents a superposition of multiple sound waves. Under the assumption that
the system is linear, it is therefore advantageous to study the response of a
system to the input $x(n)=e^{ins}$, where $s\in\mathbb{R}$ is a particular
frequency. In this case, the output is known as the frequency response of the
system. If we let $H(z)$ be the corresponding filter, then the frequency
response is expressed as

$\displaystyle y(n)$ $\displaystyle=\sum_{k=0}^{n}h(k)e^{i(n-k)s}$
$\displaystyle=\bigg{[}\sum_{k=0}^{n}h(k)e^{-iks}\bigg{]}e^{ins}$
$\displaystyle=H(e^{is})e^{ins}.$

We then see that the frequency response is bounded, with

$|y(n)|=|H(e^{is})|,\enskip n\geq 0.$

The quantity $|H(e^{is})|$ is referred to as the magnitude of the frequency
response. To get an idea of what this function looks like, consider the filter

$H(z)=\frac{0.3(z^{2}+2z+1)}{1.3z^{2}+1}.$

The poles and zeros of $H(z)$ are displayed in the following diagram:

The poles of $H(z)$ are marked with a cross and the zero of $H(z)$ is marked
with a circle. On the interval $[0,\pi]$, we therefore expect $|H(e^{is})|$ to
have a maximum around 1.5 radians and a minimum around 3.1 radians. This can
be seen in the following graph:

Often in the design of a digital filter, the goal is to develop a rational
function $H(z)$ in which the modulus satisfies a set of specifications on the
boundary $\mathbb{T}$. The effect of this would control the response of the
system to the input $x(n)=e^{ins}$. For the purpose of our discussion, we will
assume that the specifications are given in the form of a non-negative even
step function on $[-\pi,\pi]$. Such a step function is known as an ideal
digital filter. The problem of digital filter design can then be stated as
follows:

######  Problem 3.2.

For a given ideal filter $\chi(e^{is})$, find a rational function $H(z)$ with
poles inside of $\mathbb{D}$ such that $|H(e^{is})|$ is an approximation of
$\chi(e^{is})$.

Methods of approximation which guarantee stability of the filter is an
interesting topic of research. We present a method which has been modified
from the ideas in [13]. This method creates a rational function $p(z)/q(z)$,
with poles in $\mathbb{D}$, such that $|p(e^{is})|/|q(e^{is})|$ approximates
$\chi(e^{is})$ in the least-squares sense. For any $f\in L^{2}$ and any
$\eta>0$, we say that the quotient $g/h$ of two functions in $L^{2}$
approximates $f$ in the least-squares sense if

$\|hf-g\|_{L^{2}}<\eta.$

In this case, we call $g/h$ an (LS)-approximant of $f$ and write
$f\approx_{LS}g/h$.

We will present the method of approximation in three stages. In the first
stage, we will approximate the ideal filter by the magnitude of a non-
vanishing function in $H^{2}$. In the second stage, we will use optimal
polynomial approximants to approximate this non-vanishing function with the
magnitude of a rational function. In the third stage, we will alter the
numerator and denominator of the rational function in order to ensure
stability.

###  3.4. First stage of approximation

We start the first stage by defining a continuous function
$\chi_{\varepsilon}(e^{is})$ in the following way. Let
$\mathcal{S}=\\{s_{j}\\}_{j=1}^{N}$ denote the points of discontinuity of
$\chi(e^{is})$. For each $s_{j}\in\mathcal{S}$, let
$I_{j}=(s_{j}-\varepsilon/2,s_{j}+\varepsilon/2)$. Here, $\varepsilon$ is a
positive number chosen so that the intervals do not overlap and such that
$\varepsilon$ is smaller than the minimum of the step values. If
$s\notin\cup_{j=1}^{N}I_{j}$, set

$\chi_{\varepsilon}(e^{is}):=\begin{cases}\chi(e^{is})&\text{if
$\chi(e^{is})>0$}\\\ \varepsilon&\text{if $\chi(e^{is})=0$}.\end{cases}$

This creates a positive step function on
$[-\pi,\pi]\setminus\cup_{j=1}^{N}I_{j}$. Then connect each successive step
with a straight line segment. For each $s\in\cup_{j=1}^{N}I_{j}$, set
$\chi_{\varepsilon}(e^{is})$ to coincide with these segments. This creates a
non-vanishing continuous function on $[-\pi,\pi]$.

We then create an analytic function on $\mathbb{D}$ by using
$\chi_{\varepsilon}(e^{is})$. For any $z\in\mathbb{D}$, define the function

$f_{\varepsilon}(z):=\exp\left(\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{e^{is}+z}{e^{is}-z}\log\chi_{\varepsilon}(e^{is})ds\right).$

Note that $f_{\varepsilon}(z)$ is analytic in $\mathbb{D}$, non-vanishing in
$\overline{\mathbb{D}}$, and has the property that

$\log|f_{\varepsilon}(z)|=\frac{1}{2\pi}\int_{-\pi}^{\pi}\operatorname{Re}\left(\frac{e^{is}+z}{e^{is}-z}\right)\log\chi_{\varepsilon}(e^{is})ds.$

i.e., $\log|f_{\varepsilon}(z)|$ solves the Dirichlet problem in $\mathbb{D}$
with boundary values defined by $\log\chi_{\varepsilon}(e^{is})$. Therefore,
the analytic function satisfies

$|f_{\varepsilon}(e^{is})|=\chi_{\varepsilon}(e^{is})$ (8)

for all $s\in[-\pi,\pi]$. This leads us to the following theorem.

######  Theorem 3.3.

Any ideal filter $\chi$ can be approximated (in the least-squares sense) by a
non-vanishing function $f\in H^{2}$.

###### Proof.

Let $\chi(e^{is})$ be an ideal filter with a collection of discontinuities
$\mathcal{S}=\\{s_{j}\\}_{j=1}^{N}$. Given any $\eta>0$, choose $\varepsilon$
to satisfy

$0<\varepsilon<\min\bigg{\\{}\frac{\eta^{2}\pi}{N(\|\chi\|_{\infty}^{2}+1)},\frac{\eta}{\sqrt{2}}\bigg{\\}}.$

Let $\\{E_{k}\\}_{k}$ denote the collection of intervals for which
$\chi_{\varepsilon}(e^{is})=\varepsilon$. From (8), it follows that

$\displaystyle\|\chi-|f_{\varepsilon}|\|^{2}_{L^{2}}$
$\displaystyle=\|\chi-\chi_{\varepsilon}\|^{2}_{L^{2}}$
$\displaystyle=\sum_{j=1}^{N}\frac{1}{2\pi}\int_{I_{j}}|\chi(e^{is})-\chi_{\varepsilon}(e^{is})|^{2}ds$
$\displaystyle+\sum_{k}\frac{1}{2\pi}\int_{E_{k}}|\chi(e^{is})-\chi_{\varepsilon}(e^{is})|^{2}ds$
$\displaystyle\leq\varepsilon\frac{N}{2\pi}\|\chi\|_{\infty}^{2}+\varepsilon^{2}\sum_{k}\frac{1}{2\pi}\int_{E_{k}}ds$
$\displaystyle\leq\varepsilon\frac{N}{2\pi}\|\chi\|_{\infty}^{2}+\varepsilon^{2}$
$\displaystyle<\eta^{2}.$

∎

This theorem states that $|f_{\varepsilon}(z)|$ is an (LS)-approximant of
$\chi(e^{is})$. Now, since $f_{\varepsilon}(z)$ is a function in $H^{2}$ that
doesn’t vanish at the origin, the $n$-th optimal polynomial approximant
$q_{n}$ of $1/f_{\varepsilon}$ is non-vanishing on $\overline{\mathbb{D}}$
(see, e.g., Theorem 4.1). This suggests that $q_{n}$ (more specifically, the
reverse polynomial of $q_{n}$) should be a part of our rational function
$H(z)$. This observation leads to the second stage of the approximation.

###  3.5. Second stage of approximation

Given an ideal filter $\chi(e^{is})$, the first stage of the approximation
involved determining the function $f_{\varepsilon}(z)$. It then followed that
$\chi(e^{is})\approx_{LS}|f_{\varepsilon}(e^{is})|$. In the next stage, we
approximate $f_{\varepsilon}$ with the magnitude of a rational function.

Since $f_{\varepsilon}$ is actually an outer function in $H^{2}$, it follows
that $\|q_{n}f_{\varepsilon}-1\|_{L^{2}}\rightarrow 0$ as
$n\rightarrow\infty$, where $q_{n}$ denotes the $n$-th opa of
$1/f_{\varepsilon}$. Therefore, given any $\eta>0$, we can choose $N$ so that
$\|q_{N}f_{\varepsilon}-1\|_{L^{2}}<\eta.$ Moreover for any $M\geq 0$, we see
that

$\|q_{N}f_{\varepsilon}-p_{M}\|_{L^{2}}=\inf_{p\in\mathcal{P}_{M}}\|q_{N}f_{\varepsilon}-p\|_{L^{2}}<\eta,$

where $p_{M}$ denotes the orthogonal projection of $q_{N}f_{\varepsilon}$ onto
$\mathcal{P}_{M}$. Hence, we have that $p_{M}/q_{N}$ is an (LS)-approximant of
$f_{\varepsilon}$. Consequently, we have that
$|f_{\varepsilon}(e^{is})|\approx_{LS}|p_{M}(e^{is})|/|q_{N}(e^{is})|$.

It’s important to note that it’s computationally efficient to determine the
polynomials $p_{M}$ and $q_{N}$. We have already seen in Theorem 2.1 that the
coefficients of $q_{N}$ can be expressed as the solution of a system of $N+1$
linear equations, each of which are dependent only on the function
$f_{\varepsilon}$. The entries of the associated matrix $B$ can be expressed
as a Fourier coefficient of the $L^{2}$ function $|f_{\varepsilon}|^{2}$,
i.e.,

$\displaystyle
B_{jk}=\frac{1}{2\pi}\int_{-\pi}^{\pi}|f_{\varepsilon}(e^{is})|^{2}e^{i(k-j)s}ds.$

Hence, they can be computed efficiently through any available FFT algorithm.
Furthermore, $B$ is a Gram matrix generated by the vectors
$\\{z^{k}f_{\varepsilon}\\}_{k=0}^{N}$. Since these vectors are linearly
independent, it follows that $B$ is invertible. Moreover, since Gram matrices
are positive definite, and since $B$ is Hermitian and Toeplitz, we can use any
of the fast algorithms to compute its inverse. We then see that the
coefficients of $q_{N}$, say $a_{0},\dots,a_{N}$, are given by the expression

$\begin{bmatrix}a_{0}\\\ a_{1}\\\ \vdots\\\
a_{N}\end{bmatrix}=B^{-1}\begin{bmatrix}\overline{f_{\varepsilon}(0)}\\\ 0\\\
\vdots\\\ 0\end{bmatrix}.$

Hence, the coefficients are determined by the first column of $B^{-1}$ scaled
by $\overline{f_{\varepsilon}(0)}$. On the other hand, the coefficients of
$p_{M}$ are given by the first $M+1$ Fourier coefficients of
$q_{N}f_{\varepsilon}$.

###  3.6. Third stage of approximation

In the first two stages of approximation, we were able to approximate an ideal
filter $\chi(e^{is})$ with the magnitude of a rational function. More
specifically,

$\chi(e^{is})\approx_{LS}|f_{\varepsilon}(e^{is})|\approx_{LS}\frac{|p_{M}(e^{is})|}{|q_{N}(e^{is})|}.$

Since the ideal filter is assumed to be an even function on $[-\pi,\pi]$, we
have that

$\displaystyle\chi(e^{is})$ $\displaystyle=\chi(e^{-is})$
$\displaystyle\approx_{LS}|f_{\varepsilon}(e^{-is})|$
$\displaystyle\approx_{LS}\bigg{|}\frac{p_{M}(e^{-is})}{q_{N}(e^{-is})}\bigg{|}$
$\displaystyle=\bigg{|}\frac{p_{M}^{*}(e^{is})}{q_{N}^{*}(e^{is})}\bigg{|},$

where $p_{M}^{*}$ and $q_{N}^{*}$ are the reverse polynomials of $p_{M}$ and
$q_{N}$, respectively. Now, since $M$ is arbitrary, choose $M\leq N$ and
define the rational function

$H(z)=\frac{p^{*}_{M}(z)}{q^{*}_{N}(z)}.$ (9)

Then $H(z)$ is a rational function that’s analytic at $\infty$. It has the
property that $|H(e^{is})|$ is an approximation of $\chi(e^{is})$.
Furthermore, since the zeros of $q_{N}$ are outside of
$\overline{\mathbb{D}}$, and since $H(z)$ is analytic at $\infty$, it follows
that the poles of $H(z)$ are contained in the disk $\mathbb{D}$. Therefore,
the expression in (9) gives us our stable filter.

This three stage method of approximation stems from the ideas presented in
1982 by Chui and Chan in [13]. The use of opas in filter design doesn’t seem
to have gone much further in the one variable case, although some papers later
in the 80s and 90s discuss related IIR filter designs (see, e.g., [10, 11,
14]). It would be interesting to know whether opas might have some further
applications in signal processing research. There do remain some open problems
in the several variable case, as discussed in Section 4.2.

##  4\. Zeros of opas

###  4.1. One variable spaces

Let us now turn to a discussion of zeros of opas in the spaces
$H^{2}_{\omega}.$ According to Theorem 2.4, given $f\in H^{2}$, the $n$-th opa
$q_{n}$ is equal to a multiple of the $n$-th reverse orthonormal polynomial
$\varphi_{n}^{*}$. Assuming $f(0)\neq 0,$ it is well-known (see, e.g., [18,
28]) that $\varphi_{n}$ has all its zeros inside the open unit disk
$\mathbb{D},$ and therefore $\varphi_{n}^{*}$, and hence $q_{n}$, whose zeros
are reflected across the unit circle, have no zeros inside the closed unit
disk. From an engineering perspective, for a given function $f$, the produced
filter whose poles coincide with the zeros of $\varphi_{n}$ is a stable
filter. The authors of [8] studied this phenomenon for the more general spaces
$H^{2}_{\omega}$. They considered the following “minimal-zero” problem: what
is the infimum of the modulus of any zero of an optimal polynomial approximant
$q_{n}$ for any $n$ and any $f\in H^{2}_{\omega}$ (assuming $f(0)\neq 0$)? In
other words, can any zero of any opa (for any function $f$) penetrate the disk
for a given space $H^{2}_{\omega}$ and if so, how far inside the disk can that
zero appear? The authors proved the following result (see Theorem 2.1 in [8]).

######  Theorem 4.1 ([8]).

Given $f\in H^{2}_{\omega}$ such that $f(0)\neq 0,$ for each $n\in\mathbb{N},$
let $q_{n,f}$ denote the $n$-th opa of $1/f.$ Consider the extremal problem of
identifying

$M:=\inf\left\\{|z|:q_{n,f}(z)=0\mbox{ for some }f\in H^{2}_{\omega}\mbox{ and
some }n\in\mathbb{N}\right\\}.$

Then if the sequence of weights $\\{\omega_{k}\\}_{k\in N}$ is non-decreasing,
$M=1$ and the infimum is not achieved. On the other hand, if the sequence of
weights is such that there exist $k,n\in\mathbb{N}$ that satisfy
$\omega_{k+n+1}<\omega_{k+1}/4,$ then $M<1$ and the infimum is a minimum.

In particular, for any of the Dirichlet-type spaces $D_{\alpha},$ if
$\alpha\geq 0$ (for example, in $H^{2}$ or in the classical Dirichlet space)
the infimum is $1$, and all the zeros of the optimal approximants lie outside
the closed unit disk. On the other hand, if $\alpha<0,$ there exist functions
$f\in D_{\alpha}$ whose optimal polynomial approximants have zeros inside the
unit disk. In particular, in the Bergman space $A^{2},$ the authors showed
(see Theorem 5.1 of [8] that $M=2\sqrt{2}/3$ and the corresponding extremal
function is $f(z)=\frac{1}{(1-z/\sqrt{2})^{3}}.$

The proof of the existence of an extremal in the case that the weights satisfy
the required inequality in Theorem 4.1 from [8] is difficult and relies on
machinery from the theory of orthogonal polynomials. Below, we give an outline
of the argument involved in the easier parts of the proof and only mention the
general ideas in the proof of existence of the extremal.

###### Proof.

First note that it is enough to consider $n=1$, since if $z_{1}$ is a zero of
some optimal approximant $q_{n}$ ($n>1$) for some function $f$, then since

$\|q_{n}f-1\|_{\omega}=\|(z-z_{1})\frac{q_{n}}{z-z_{1}}f-1\|_{\omega},$

$z_{1}$ is a zero of a first order approximant for some function that is a
multiple of $\frac{q_{n}}{z-z_{1}}f.$

In that case, if $z_{1}$ is a zero of a first order approximant for some
function $f$, it is not hard to see that

$z_{1}=\frac{\|zf\|_{\omega}^{2}}{\langle f,zf\rangle_{\omega}},$

and therefore the extremal problem becomes to find

$M=\inf\left\\{\frac{\|zf\|_{\omega}^{2}}{|\langle f,zf\rangle_{\omega}|},f\in
H^{2}_{\omega}\right\\}.$ (10)

Using the Cauchy-Schwarz inequality and noticing that if the sequence
$\\{\omega_{k}\\}_{k\in N}$ is non-decreasing, then
$\|zf\|_{\omega}\geq\|f\|_{\omega}$ gives that for those weights, $M\geq 1.$
Moreover, the condition on the weights given in the introduction ensure that
$M\leq 1.$ Therefore $M=1$ in that case, and it is not hard to see that
equality cannot hold in the Cauchy-Schwarz inequality, which implies that the
infimum is not attained.

On the other hand, if the weights satisfy the condition that
$\omega_{k+n+1}<\omega_{k+1}/4,$ the authors of [8] show that if

$f(z)=z^{k}T_{n}\left(\frac{1+z}{1-z}\right),$

where $T_{n}(g)$ is the $n$-th Taylor polynomial of a function $g$, then the
ratio

$\frac{\|zf\|^{2}_{\omega}}{|\langle f,zf\rangle_{\omega}|}$

is strictly less than $1$, by explicit computation.

The remainder of the proof involves reducing the extremal problem (10) to one
that only involves functions $f$ that are polynomials with positive
coefficients and of degree at most $N$. If $Q_{N}$ is the corresponding
extremal polyomial, the authors then show that the coefficients of these
extremal polynomials $Q_{N}$ satisfy a three-term recurrence relation, and are
therefore orthogonal polynomials on the real line, which in turn are connected
to Jacobi matrices and associated homogeneous linear differential equations.
The authors show that if $J$ is the Jacobi matrix with entries
$J_{ij}=\sqrt{\frac{\omega_{j}}{\omega_{j+1}}}$ if $|i-j|=1$ and $J_{ij}=0$
otherwise, then $M=\frac{2}{\|J\|},$ where $\|J\|$ is the norm of the matrix
$J.$ The authors identify the extremal function in terms of certain
orthonormal polynomials associated with $Q_{N}$ and with the norm of the
matrix $J$ (see Corollary 4.5 of [8]). ∎

In general, it is very difficult to calculate $M$ explicitly, and the authors
investigate this question in detail, in particular in some other spaces
involving integral norms, but in particular, the following question remains
open.

######  Open Question 1.

What is $M$ for the spaces $D_{\alpha}$ when $\alpha<0,$ $\alpha\neq-1?$ What
is the corresponding extremal function? Given the connection with Jacobi
matrices above, this question can be rephrased as, for $\alpha<0,$
$\alpha\neq-1,$ what is the norm of the Jacobi matrix $J$ defined by
$J_{ij}=\sqrt{\frac{(j+1)^{\alpha}}{(j+2)^{\alpha}}}$ if $|i-j|=1$ and
$J_{ij}=0$ otherwise?

One may also ask the corresponding question for a _fixed_ integer $n>1.$ In
addition, there are many interesting questions surrounding the extremal
function, which satisfies a differential equation and is equal to the
derivative of the reproducing kernel evaluated at a particular point of the
disk, for certain spaces, which seems to warrant further investigation (see
Section 8 of [8]).

Another direction of inquiry investigates the limit points of the zeros of the
optimal polynomial approximants. Indeed, it turns out that for a general class
of weighted spaces where the weights are associated with certain “regular
measures”, and for (say) cyclic functions $f$ such that $1/f$ has a
singularity on the unit circle, every point of the unit circle is a limit
point of the zeros of the optimal polynomial approximants of $1/f.$ This is an
analogue of a beautiful theorem of Jentzsch that states that if an analytic
function in the unit disk has radius of convergence $1$, then every point on
the unit circle is a limit point of the zeros of the Taylor polynomials of
that function. The proof of the theorem follows the outline of the original
proof of Jentzsch’s theorem, but knowledge of the precise asymptotic behavior
of the orthogonal polynomials for regular measures connected to the opas is
required. For details, see [8, Theorem 6.2]. In fact, the authors prove that
asymptotically, for each $\varepsilon>0,$ the zeros of the $n$-th opa for such
a function $f$ lie in a disk of radius $1+\varepsilon.$ (See Theorem 6.1 in
[8].) However, more detail on the precise behavior of the zeros is needed. For
instance, the following question is open.

######  Open Question 2.

Given a cyclic function $f\in H^{2}_{\omega},$ and given $\zeta\in\mathbb{T},$
is there a sector with vertex at $\zeta$ that is devoid of zeros of opas? More
generally, are there regions or rays that the zeros avoid?

Finally, another line of investigation related to zeros involves cyclicity. In
[5, Theorem 6.1], the authors find a characterization of cyclicity of a
function $f\in H^{2}$ based on the relationship between the zeros of the opas
and the value of $f$ at the origin. Since cyclic functions in $H^{2}$ are
outer functions and are well-understood, this characterization is not that
useful for $H^{2},$ but perhaps this idea can be extended to other spaces.
Indeed, one might hope that such a characterization would give insight into
the Brown and Shields Conjecture. Thus, although vague, the following is open.

######  Open Question 3.

For a given space $H^{2}_{\omega}$ (or, say for the Bergman space $A^{2}$ or
the classical Dirichlet space $\mathcal{D}$), is there a characterization of a
cyclic function $f$ that relies on the behavior of the zeros of the opas for
$1/f$?

###  4.2. Several variable spaces and Shanks Conjecture

When considering Hilbert spaces of analytic functions of several variables,
the zeros of opas are less well-understood, partly because zeros of functions
of several variables are no longer isolated. On the other hand, this topic was
of great interest in the engineering literature of the 70s in connection with
filters, as discussed earlier. Two recent papers (see [25, 26]) discuss the
several variable situation in detail, so we just mention a few relevant items
of interest here.

For simplicity, let us consider the Dirichlet-type spaces of the bidisk
defined as follows. Let $(\alpha_{1},\alpha_{2})\in\mathbb{R}^{2}.$ Given

$f(z_{1},z_{2})=\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}a_{j,k}z_{1}^{j}z_{2}^{k}$

analytic in $\mathbb{D}^{2},$ we will say
$f\in\mathcal{D}_{\alpha_{1},\alpha_{2}}$ if

$\|f\|_{\alpha_{1},\alpha_{2}}^{2}:=\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}(j+1)^{\alpha_{1}}(k+1)^{\alpha_{2}}|a_{j,k}|^{2}<\infty.$

This makes $\mathcal{D}_{\alpha_{1},\alpha_{2}}$ into a reproducing kernel
Hilbert space. In order to define optimal polynomial approximants, as the
authors of [25] note, one needs to order the monomials $z_{1}^{j}z_{2}^{k}$ in
some fashion. Let us assume that we have chosen such an ordering (for
instance, the degree lexicographic ordering), and let $\mathcal{P}_{n}$ be the
span of the first $n+1$ monomials. In this way, given
$f\in\mathcal{D}_{\alpha_{1},\alpha_{2}},$ not identically $0$, we can define,
as before, the $n$-th optimal polynomial approximant of $1/f$ in
$\mathcal{D}_{\alpha_{1},\alpha_{2}}$ to be the polynomial $q_{n}$ that
minimizes $\|pf-1\|_{\alpha_{1},\alpha_{2}}$ among all polynomials
$p\in\mathcal{P}_{n}$.

In [2], the authors extended their results from [1] to the case when
$\alpha_{1}=\alpha_{2}=\alpha$ to obtain rates of decay for
$\|q_{n}f-1\|_{\alpha,\alpha}$ for functions $f$ analytic in the closed unit
bidisk with no zeros in the bidisk. They also gave examples of polynomials
with no zeros on the bidisk that are not cyclic in
$\mathcal{D}_{\alpha,\alpha}$ for $\alpha>1/2$. Cyclic polynomials in these
Dirichlet spaces of the bidisk (for $\alpha_{1}=\alpha_{2}$) were completely
characterized in [6]. In [21], the authors characterized cyclic polynomials in
the anisotropic Dirichlet spaces, that is, the more general case of
$\mathcal{D}_{\alpha_{1},\alpha_{2}}$ when $\alpha_{1}$ and $\alpha_{2}$ may
be different.

In [25], the authors discuss the history of the problem of using optimal
polynomial approximants to design digital filters of two variables. In this
quest, in [27], the authors conjectured that if $f$ is a polynomial that is
zero-free in the bidisk, then optimal polynomial approximants of $1/f$ in the
Hardy space of the bidisk would also be zero-free in the bidisk. This became
known as Shanks Conjecture, which was disproved in [17]. A simplified counter-
example can be found in [25]. However, the following weaker version of the
Shanks conjecture remains open.

######  Open Question 4.

[Weak Shanks Conjecture] Let $f(z_{1},z_{2})$ be a polynomial that doesn’t
vanish in $\overline{\mathbb{D}^{2}}$. Then the optimal polynomial approximant
of $1/f$ in $H^{2}(\mathbb{D}^{2})$ is zero-free in $\mathbb{D}^{2}$.

It’s interesting to know that although this result is unknown in
$H^{2}(\mathbb{D}^{2})$, it does fail in other function spaces of the bidisk,
including the Bergman space $A^{2}(\mathbb{D}^{2})$ (see Example 22 of [25]).

##  5\. Convergence Results

One set of questions of great interest but with little known so far is about
rate of convergence of opas for a given cyclic function $f$. Since $f$ is
cyclic, we know that $\|q_{n}f-1\|_{\omega}$ approaches $0$ at the fastest
possible rate in terms of the degree. What is that rate of convergence, for a
given $f$? In addition, since norm convergence implies uniform convergence on
compact subsets of the open unit disk, we know that $q_{n}f$ approaches $1$
pointwise, uniformly on compact subsets of $\mathbb{D},$ but what more can be
said about convergence on the circle? In what follows, we discuss what is
known about these questions and some related open problems.

###  5.1. Norm Convergence

In [1], the authors studied the rate of decay of $\|q_{n}f-1\|_{\alpha}$ for
certain simple cyclic functions $f$ in the Dirichlet-type spaces $D_{\alpha}$
for $\alpha\leq 1.$ Note that for $\alpha>1,$ a function is cyclic if and only
if it does not vanish in the closed disk. Also, if $f$ is such that $1/f$ is
analytic in the closed disk, then it is easy to see that the rate of decay of
$\|q_{n}f-1\|_{\alpha}$ is exponential. Therefore the question of rate of
decay is most interesting for $\alpha\leq 1$ and for functions $f$ such that
$1/f$ has a singularity on the unit circle. The simplest possible function to
consider then is $f(z)=1-z.$

For the function $1-z,$ as we saw in Example 2.2, an explicit formula for
$q_{n}$ can be obtained, and moreover, the authors of [1] calculated the rate
of decay of $\|q_{n}f-1\|_{\alpha}$ and showed that this rate of decay does
not change for _any_ polynomial whose zeros are outside the unit disk (with at
least one zero on the circle). They then extended that result to any function
that admits an analytic extension to the closed unit disk. More specifically,
their result is the following.

######  Theorem 5.1.

Let $\alpha\leq 1$ and let $f$ be analytic in the closed unit disk and have
zeros outside $\mathbb{D}.$ Then for each $n\in\mathbb{N}$ there exists a
constant $C$ independent of $n$ such that

$\|q_{n}f-1\|^{2}_{\alpha}\leq\begin{cases}\frac{C}{(n+1)^{1-\alpha}}\,\,\mbox{for
}\alpha<1\\\ \frac{C}{\log^{+}(n+1)}\,\,\mbox{for }\alpha=1.\end{cases}$

Moreover, if $f$ has a zero on the unit circle, this rate of convergence is
sharp.

In [16], the authors showed that the rate of decay of
$\|q_{n}f-1\|_{\omega}^{2}$ for $f(z)=1-z$ is precisely equal to
$\left(\frac{1}{\sum_{k=0}^{n+1}\frac{1}{\omega_{k}}}\right)$; the authors of
[25] then exploited that single-variable rate to identify rates of decay for
various analogues of $1-z$ in some of the several variable spaces. In [22],
the authors came up with an example of a lacunary series of the type
$f(z)=1+\sum_{k=1}^{\infty}a_{k}z^{2^{k}}$ whose corresponding rate of decay
in the Dirichlet-type spaces $\mathcal{D}_{\alpha}$ is _slower_ than the rate
in Theorem 5.1. For instance, they construct a lacunary function $f$ in
$H^{2}$ such that $1/(\log n)^{1+\varepsilon}$ is a lower bound for the rate
of decay of $\|q_{n}f-1\|_{H^{2}}^{2}$. However, for most other functions, the
rate of decay of $\|q_{n}f-1\|_{\alpha}$ is unknown, and thus, the following
general problem is essentially open for most functions.

######  Open Question 5.

Given a function $f\in D_{\alpha}$ such that $f$ does not have an analytic
extension to the closed disk, find the rate of decay of
$\|q_{n}f-1\|_{\alpha}$.

In [12], the author considered a slightly different convergence question in
the Hardy space $H^{2}$ related to double least squares. Given a _polynomial_
$f$ of degree $n$, for each $k\in\mathbb{N},$ let $q_{k}$ be the $k$-th
optimal polynomial approximant (in $H^{2}$) of $1/f.$ Then let $Q_{n,k}$ be
the $n$-th optimal polynomial approximant of $1/q_{k}.$ Since we expect
$q_{k}$ to be some kind of approximation of $1/f$ and $Q_{n,k}$ some kind of
approximation of $1/q_{k},$ we expect $Q_{n,k}$ to approximate $f$ as
$k\rightarrow\infty.$ Indeed, this will be the case (as shown in [12]) for
cyclic polynomials $f$ (i.e., ones that have no zeros in $\mathbb{D}.$) In
fact, Chui showed that the limit of the polynomials $Q_{n,k}$ is a polynomial
of degree $n$ that preserves the zeros of $f$ that are outside $\mathbb{D}$
and contains in addition the reflection of the zeros of $f$ that are inside
$\mathbb{D}.$ We summarize these two results of Chui in the following theorem.

######  Theorem 5.2 ([12],Theorems 2.1 and 3.1).

Let $f(z)=p(z)\cdot\prod_{j=1}^{m}(\alpha_{j}-z)$ be a polynomial of degree
$n$ where $\alpha_{j}\in\mathbb{D}\setminus\\{0\\}$ and $p$ is a polynomial of
degree $n-m$ that has no zeros in $\mathbb{D}.$ Let $q_{k}$ be the $k$-th
optimal polynomial approximant in $H^{2}$ of $1/f,$ and let $Q_{n,k}$ be the
$n$-th optimal polynomial approximant of $1/q_{k}.$ Then as
$k\rightarrow\infty,$ $\|Q_{n,k}-\tilde{f}\|_{H^{2}}\rightarrow 0,$ where

$\tilde{f}(z)=p(z)\cdot\prod_{j=1}^{m}(1/\bar{\alpha_{j}}-z).$

As a consequence, $\|Q_{n,k}-f\|_{H^{2}}\rightarrow 0$ if and only if $f$ is a
cyclic polynomial, i.e., has no zeros in $\mathbb{D}.$

Izumino (see [20]) extended Chui’s results to functions $f\in H^{\infty}$ such
that $1/f\in H^{\infty}$ using operator theory methods. He also proved a
conjecture that Chui had considered related to double least squares of _lower_
degree than the degree of the original polynomial $f$. More specifically, he
proved the following.

######  Theorem 5.3 (Theorem 3.5 of [20]).

Let $f(z)=p(z)\cdot\prod_{j=1}^{m}(\alpha_{j}-z),$ where $|\alpha_{j}|=1$ for
$j=1,\ldots,m$ and $p\in H^{\infty}$ is an outer function. Then for each
$n=0,\ldots,m-1,$ $\|Q_{n,k}\|_{H^{2}}\rightarrow 0$ as $k\rightarrow\infty.$

Notice that the convergence result of Chui’s is a result in a finite
dimensional space, the space of polynomials of degree at most $n,$ and
therefore convergence of the _double_ least-squares also happens for instance
uniformly in the closed disk. In general, though, if $f$ is a cyclic function,
we know $(q_{n}f-1)(z)$ converges to $0$ on compact subsets of the open unit
disk, because of the norm convergence, but what happens on the unit circle?
For a “good” function, should we expect that $q_{n}(\zeta)\rightarrow
1/f(\zeta)$ for $\zeta\in\mathbb{T}$? This leads to another series of
questions that concern pointwise convergence of optimal polynomial
approximants on the unit circle, which we discuss in the next section.

###  5.2. Pointwise convergence

The boundary behavior of optimal polynomial approximants depends heavily on
the function $f$ whose inverse they are approximating. In two recent papers
([4, 7]), the authors investigate two distinct phenomena. In [4], the authors
show that there exist many $H^{2}$ functions $f$ whose opas on the unit circle
can approximate any complex number. This is a kind of universality phenomenon.
More precisely, they proved a theorem that implies the following.

######  Theorem 5.4 ([4]).

For any $\zeta\in\mathbb{T},$ there exists a $G_{\delta}$-dense set of
functions $f$ in $H^{2}$ (or in the classical Dirichlet space) with
corresponding optimal approximants $q_{n}$ such that the set of points
$\\{q_{n}(\zeta)\\}$ is dense in $\mathbb{C}.$

Thus clearly it is not the case that $q_{n}(\zeta)$ converges to $f(\zeta)$
for that particular point and for that large set of functions $f$. However, if
$f$ is well-behaved enough, pointwise convergence of the opas at points on the
circle that are not zeros of $f$ does occur. In fact, even more is true, as
seen in the following.

######  Theorem 5.5 ([7]).

Let $f$ be a polynomial with simple zeros that lie outside the open unit disk,
and let $q_{n}$ be the $n$-th opa of $1/f$ in the Hardy space $H^{2}$ or the
Bergman space $A^{2}.$ Then $1-q_{n}f$ converges to $0$ uniformly on compact
subsets of $\overline{\mathbb{D}}\setminus Z(f)$, where $Z(f)$ is the set of
zeros of $f$.

The proof of this result is somewhat technical and relies on estimates of the
coefficients of $1-q_{n}f$ which are tractable in the Hardy and the Bergman
space. It seems likely based on preliminary work that the result holds for any
polynomial with no zeros in $\mathbb{D}$, but the computations become more
complicated. It would be interesting to know whether these results still hold
for more general spaces and how “good” the functions have to be in order to
get convergence versus the universality behavior observed in Theorem 5.4. Thus
we conclude with the following open questions.

######  Open Question 6.

Suppose $f$ is a polynomial with zeros $Z(f)$ in the complement of
$\mathbb{D}$ and let $q_{n}$ be the $n$-th opa of $1/f$ in $H^{2}_{\omega}.$
For which weights $\\{\omega_{k}\\}_{k\in\mathbb{N}}$ is it true that
$1-q_{n}f$ converges to $0$ uniformly on compact subsets of
$\overline{\mathbb{D}}\setminus Z(f)$?

######  Open Question 7.

Does Theorem 5.5 hold for functions other than polynomials, and if so, for
which ones? On the other hand, can we characterize the functions for which the
universality behavior as in Theorem 5.4 occurs?

## References

  * [1] Bénéteau, C., Condori, A., Liaw, C., Seco, D., and Sola, A., Cyclicity in Dirichlet-type spaces and extremal polynomials, _J. Anal. Math._ 126 (2015) 259–286.
  * [2] Bénéteau, C., Condori, A., Liaw, C., Seco, D., and Sola, A., Cyclicity in Dirichlet-type spaces and extremal polynomials II: Functions on the bidisk, _Pacific J. Math._ 276 (2015) no. 1, 35-58.
  * [3] Bénéteau, C., Fleeman, M., Khavinson, D., Seco, D., and Sola, A., Remarks on inner functions and optimal approximants, _Canad. Math. Bull._ 61 (2018) 704–716.
  * [4] Bénéteau, C., Ivrii, O., Manolaki, M., and Seco, D., Simultaneous zero-free approximation and universal optimal polynomial approximants, _J. Approx. Theory_ 256 (2020), 105389, 13 pp.
  * [5] Bénéteau, C., Khavinson, D., Liaw, C., Seco, D., and Sola, A., Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants, _J. London Math. Soc._ 94 (2016), no. 3, 726–746.
  * [6] Bénéteau, C., Knese, G., Kosiński, Ł., Liaw, C., Seco, D., and Sola, A., Cyclic polynomials in two variables, _Trans. Am. Math. Soc._ 368 (2016), 8737–8754.
  * [7] Bénéteau, C., Manolaki, M., and Seco, D., Boundary behavior of optimal polynomial approximants, _Constr. Approx._ , 2020, https://link.springer.com/article/10.1007/s00365-020-09508-z.
  * [8] Bénéteau, C., Khavinson, D., Liaw, C., Seco, D., and Simanek, B., Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems, _Rev. Mat. Iber._ 35 (2019), no. 2, 607–642.
  * [9] Brown, L. and Shields, A., Cyclic vectors in the Dirichlet space, _Trans. Amer. Math. Soc._ 285 (1984), 269–304.
  * [10] Chen, B-S., Chiou, B-W., and Peng, S-C., Minimum sensitivity IIR filter design using principal component approach, _IEE Proceedings G (Circuits, Devices and Systems)_ 138 (1991), no. 4, 474–482.
  * [11] Chen, B-S., Peng, S-C., and Chiou, B-W., IIR filter design via optimal Hankel-norm approximation, _IEE Proceedings G (Circuits, Devices and Systems)_ 139 (1992), no. 5, 586–590.
  * [12] Chui, C., Approximation by double least-squares inverses, _J. Math. Anal. Appl._ 75 (1980), 149–163.
  * [13] Chui, C. and Chan, A., Application of approximation theory methods to recursive digital filter design, _IEEE Transactions on acoustics, speech, and signal processing_ ASSP-30, No. 1 (1982), 18–24.
  * [14] Durrani, T., and Chapman, R., Optimal all-pole filter design based on discrete prolate spheroidal sequences, _IEEE transactions on acoustics, speech, and signal processing_ 32 (1984), no. 4, 716–721.
  * [15] Felder, C. General Optimal Polynomial Approximants, Stabilization, and Projections of Unity, arXiv:2003.10015.
  * [16] Fricain, E., Mashreghi, J., and Seco, D., Cyclicity in Reproducing Kernel Hilbert Spaces of analytic functions, _Comput. Methods Funct. Theory_ (2014) Issue 14, 665-680.
  * [17] Genin, Y. and Kamp, Y., Counter-examples in the best least-squares inverse stabilization of 2D-recursive filters, _Elec. Lett._ 11 (1975), 330–331.
  * [18] Geronimus, Ya. L., Orthogonal polynomials: estimates, asymptotic formulas, and series of polynomials orthogonal on the unit circle and on an interval, Authorized translation from the Russian, Consultants Bureau, New York, 1961.
  * [19] I.F. Gonos, L.I. Virirakis, N.E. Mastorakis, and M.N.S. Swamy, Evolutionary design of 2-dimensional recursive filters via the computer language GENETICA, _IEEE Transactions on Circuits and Systems II_ , 53, no.4 (2006), 254–258.
  * [20] Izumino, S., Generalized inverses of Toeplitz operators and inverse approximation in $H^{2},$ _Tohoku J. Math_ (2) 37 (1985), 95–99.
  * [21] Knese, G., Kosiński, Ł., Ransford, T. and Sola, A., Cyclic polynomials in anisotropic Dirichlet spaces, _J. Anal. Math._ 138, 23–47 (2019).
  * [22] Richter, S., Sola, A., and Sundberg, C., private communication.
  * [23] Robinson, E., Structural properties of stationary stochastic processes with applications, in _Time Series Analysis_ , Wiley, New York, 1963.
  * [24] Seco, D. and Téllez, R., Polynomial approach to cyclicity for weighted $\ell^{p}_{A}$ spaces, arXiv:200102130.
  * [25] Sargent, M. and Sola, A., Optimal approximants and orthogonal polynomials in several variables, arXiv:2002.08790.
  * [26] Sargent, M. and Sola, A., Optimal approximants and orthogonal polynomials in several variables II: families of polynomials in the unit ball, arXiv:2009.01793.
  * [27] Shanks, J., Treitel, S., and Justice, J., Stability and synthesis of two-dimensional recursive filters, _IEEE Trans. on Audio and Electroacoustics AU-20_ , 2 (1972), 115–128.
  * [28] Simon, B., Orthogonal polynomials on the unit circle, Part I: Classical Theory, AMS Colloquium Publications 53, Amer. Math. Soc., Providence, RI, 2005.