We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.
We obtain closed expressions for weighted orthogonal polynomials and optimal approximants associated with the function $f(z)=1-\frac{1}{\sqrt{2}}(z_1+z_2)$ and a scale of Hilbert function spaces in the unit $2$-ball having reproducing kernel $(1-\langle z,w\rangle)^{-γ}$, $γ>0$. Our arguments are elementary but do not rely on reduction to the one-dimensional case.
Given a system of functions $\textup{\textbf{F}}=(F_1,\ldots,F_d),$ analytic on a neighborhood of some compact subset $E$ of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in terms of the orthogonal expansions of the components $F_k, k=1,\ldots,d,$ with respect to a sequence of orthonormal polynomials associated with a measure $μ$ whose support is contained in $E$. Such sequences of vector rational functions resemble row sequences of type II Hermite-Padé approximants. Under appropriate assumptions on $μ,$ we give ne…
The study of inner and cyclic functions in $\ell^p_A$ spaces requires a better understanding of the zeros of the so-called optimal polynomial approximants. We determine that a point of the complex plane is the zero of an optimal polynomial approximant for some element of $\ell^p_A$ if and only if it lies outside of a closed disk (centered at the origin) of a particular radius which depends on the value of $p$. We find the value of this radius for $p\neq 2$. In addition, for each positive integer $d$ there is a polynomial $f_d$ of degree at most $d$ that minimizes the modulus of the root of its…
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…
In the paper, we propose two new conjectures about the convergence of Hermite Approximants of multivalued analytic functions of Laguerre class ${\mathscr L}$. The conjectures are based in part on the numerical experiments, made recently by the authors in [26] and [27].
Let $E$ be a closed subset of the unit circle of measure zero. Recently, Beise and Müller showed the existence of a function in the Hardy space $H^2$ for which the partial sums of its Taylor series approximate any continuous function on $E$. In this paper, we establish an analogue of this result in a non-linear setting where we consider optimal polynomial approximants of reciprocals of functions in $H^2$ instead of Taylor polynomials. The proof uses a new result on simultaneous zero-free approximation of independent interest. Our results extend to Dirichlet-type spaces $\mathcal{D}_α$ for $α\i…
We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions.
We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzsch-type theorems describing where they accumulate. As a consequence, we obtain detailed information regarding zeros of reproducing kernels in weighted spaces of analytic functions.
In this paper, we provide an efficient method for computing the Taylor coefficients of $1-p_n f$, where $p_n$ denotes the optimal polynomial approximant of degree $n$ to $1/f$ in a Hilbert space $H^2_ω$ of analytic functions over the unit disc $\mathbb{D}$, and $f$ is a polynomial of degree $d$ with $d$ simple zeros. As a consequence, we show that in many of the spaces $H^2_ω$, the sequence $\{1-p_nf\}_{n\in \mathbb{N}}$ is uniformly bounded on the closed unit disc and, if $f$ has no zeros inside $\mathbb{D}$, the sequence $\{1-p_nf \}$ converges uniformly to 0 on compact subsets of the comple…
Over the past several years, optimal polynomial approximants (OPAs) have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the properties of their zeros. In this paper, we introduce the notion of optimal polynomial approximant in the space $L^p$, $1\leq p\leq\infty$. We begin our treatment by showing existence and uniqueness for $1<p<\infty$. For the extreme cases of $p=1$ and $p=\infty$, we show that uniqueness does not necessarily hold. We continue our development by elaborating on the special case of $L^2$. Here, we create a tes…
We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$ on a compact set $K$, the critical points of our approximants may be taken to lie in any given domain containing $K$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$.
In previous works, an approach to the study of cyclic functions in reproducing kernel Hilbert spaces has been presented, based on the study of so called \emph{optimal polynomial approximants}. In the present article, we extend such approach to the (non-Hilbert) case of spaces of analytic functions whose Taylor coefficients are in $\ell^p(ω)$, for some weight $ω$. When $ω=\{(k+1)^α\}_{k\in \mathbb{N}}$, for a fixed $α\in \mathbb{R}$, we derive a characterization of the cyclicity of polynomial functions and, when $1<p<\infty$, we obtain sharp rates of convergence of the optimal norms.
In the study of the cyclicity of a function $f$ in reproducing kernel Hilbert spaces an important role is played by sequences of polynomials $\{p_n\}_{n\in \mathbb{N}}$ called \emph{optimal polynomial approximants} (o.p.a.). For many such spaces and when the functions $f$ generating those o.p.a. are polynomials without zeros inside the disk but with some zeros on its boundary, we find that the weakly asympotic distribution of the zeros of $1-p_nf$ is the uniform measure on the unit circle.
We study the logarihtnmic asymptotic of multiple orthogonal polynomials arising in a mixed type Hermite-Padé approximation problem associated with the rational perturbation of a Nikishin system of functions. The formulas obtained allow to give exact estimates of the rate of convergence of the corresponding Hermite-Padé approximants.
In the context of orthogonal polynomials in the plane we introduce the notion of a polynomially small (PS) perturbation of a measure. In such a case we establish relative asymptotic results for the two sequences of the associated orthonormal polynomials. We also provide results dealing with the behaviour of the zeros of PS perturbations of area orthogonal (Bergman) polynomials.
We give the strong asymptotic of Cauchy biorthogonal polynomials under the assumption that the defining measures are supported on non intersecting intervals of the real line and satisfy Szegő's condition. The biorthogonal polynomials are connected with certain mixed type Hermite-Padé polynomials, which verify full orthogonality relations with respect to some varying measures. Thus, the strong asymptotic of orthogonal polynomials with respect to varying measures plays a key role in the study.
In this paper we study the asymptotic analysis of the orthogonal trigonometric polynomials by the Riemann-Hilbert problem for the periodic analytic functions.
