arXiv subjects

Jeffrey S. Geronimo

Publications and source records attributed to Jeffrey S. Geronimo.

At least 19 recordsLinked to original sources

arXiv2026

Compactly supported, orthogonal, continuous piecewise polynomial multiresolution analysis

We present explicit representations in terms of hypergeometric functions for the scaling functions in the $C^0$ orthogonal multiresolution analyses associated with piecewise continuous polynomials. Closed formulas for the Mellin transform of these functions as well as their Fourier transforms are derived. Some new multiresolution analyses whose scaling functions have coefficients that are rational numbers are introduced and discussed.

math.CA↗

arXiv2026

Bernstein-Szegő measures in the plane

We define a class of Bernstein-Szegő measures on $\mathbb{R}^2$ and we establish their spectral properties, providing a natural extension of the one-dimensional theory. We also derive conditions involving finitely many moments, which are new in the two-dimensional setting, and which completely characterize these measures. A key ingredient in the theory on the real line stems from the fact that a measure $μ$ on $\mathbb{R}$ determines a unique sequence of orthonormal polynomials which gives a simple formula for $dμ/dx $ in the Bernstein-Szegő family. Since there is no canonical way to introduce orthonormal polynomials in the plane, our extension is based on a new identity which connects a Fejér-Riesz factorization of the weight to a polynomial depending on three variables associated with $μ$. Using recent results in the bivariate trigonometric Fejér-Riesz factorization problem, we define a nontrivial two-dimensional extension of the Szegő mapping which provides explicit orthonormal bases of the spaces associated with Bernstein-Szegő measures on $\mathbb{R}^2$. An important part of the paper is devoted to a self-contained development of the Bernstein-Szegő theory for matrix-valued functionals. The proofs combine techniques from real analysis, complex analysis and algebra.

math.CA↗

arXiv2025

A Fourier analysis approach to disprove the weak Shanks conjecture

We derive formulas for the Fourier coefficients of $|f|^2$, where $f(z_1,z_2)=(1-\frac{z_1+z_2}{r})^{-α}$, in terms of hypergeometric functions. Using these formulas we provide additional counterexamples to the weak Shanks conjecture, which was recently disproven by Bénéteau, Khavinson and Seco. The obtained formulas allow for (numerical) optimization over the parameters $α$ and $r$.

math.CV↗

arXiv2021

The Fourier extension method and discrete orthogonal polynomials on an arc of the circle

The Fourier extension method, also known as the Fourier continuation method, is a method for approximating non-periodic functions on an interval using truncated Fourier series with period larger than the interval on which the function is defined. When the function being approximated is known at only finitely many points, the approximation is constructed as a projection based on this discrete set of points. In this paper we address the issue of estimating the absolute error in the approximation. The error can be expressed in terms of a system of discrete orthogonal polynomials on an arc of the unit circle, and these polynomials are then evaluated asymptotically using Riemann--Hilbert methods.

math.NA↗

arXiv2021

Connection coefficients for ultraspherical polynomials with argument doubling and generalized bispectrality

We start by presenting a generalization of a discrete wave equation that is particularly satisfied by the entries of the matrix coefficients of the refinement equation corresponding to the multiresolution analysis of Alpert. The entries are in fact functions of two discrete variables and they can be expressed in terms of the Legendre polynomials. Next, we generalize these functions to the case of the ultraspherical polynomials and show that these new functions obey two generalized eigenvalue problems in each of the two discrete variables, which constitute a generalized bispectral problem. At the end, we make some connections to other problems.

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arXiv2021

The autoregressive filter problem for multivariable degree one symmetric polynomials

The multivariable autoregressive filter problem asks for a polynomial $p(z)=p(z_1, \ldots , z_d)$ without roots in the closed $d$-disk based on prescribed Fourier coefficients of its spectral density function $1/|p(z)|^2$. The conditions derived in this paper for the construction of a degree one symmetric polynomial reveal a major divide between the case of at most two variables vs. the the case of three or more variables. The latter involves multivariable elliptic functions, while the former (due to [J. S. Geronimo and H. J. Woerdeman, Ann. of Math. (2), 160(3):839--906, 2004]) only involve polynomials. The three variable case is treated with more detail, and entails hypergeometric functions. Along the way, we identify a seemingly new relation between $_2F_1(\frac13,\frac23;1;z)$ and $_2F_1(\frac12,\frac12;1;\widetilde{z})$.

math.CA↗

arXiv2020

Spectral density functions of bivariable stable polynomials

The relationship between a stable multivariable polynomial $p(z)$ and the Fourier coefficients of its spectral density function $1/|p(z)|^2$, is further investigated. In this paper we focus on the radial asymptotics of the Fourier coefficients for a specific choice of a two variable polynomial. Hypergeometric functions appear in the analysis, and new results are derived for these as well.

math.CA↗

arXiv2016

Alpert multiwavelets and Legendre-Angelesco multiple orthogonal polynomials

We show that the multiwavelets, introduced by Alpert in 1993, are related to type I Legendre-Angelesco multiple orthogonal polynomials. We give explicit formulas for these Legendre-Angelesco polynomials and for the Alpert multiwavelets. The multiresolution analysis can be done entirely using Legendre polynomials, and we give an algorithm, using Cholesky factorization, to compute the multiwavelets and a method, using the Jacobi matrix for Legendre polynomials, to compute the matrices in the scaling relation for any size of the multiplicity of the multiwavelets.

math.CA↗

arXiv2015

Bernstein-Szegő measures, Banach algebras, and scattering theory

We give a simple and explicit description of the Bernstein-Szego type measures associated with Jacobi matrices which differ from the Jacobi matrix of the Chebyshev measure in finitely many entries. We also introduce a class of measures M which parametrizes the Jacobi matrices with exponential decay and for each element in M we define a scattering function. Using Banach algebras associated with increasing Beurling weights, we prove that the exponential decay of the coefficients in a Jacobi matrix is completely determined by the decay of the negative Fourier coefficients of the scattering function. Combining this result with the Bernstein-Szego type measures we provide different characterizations of the rate of decay of the entries of the Jacobi matrices for measures in M.

math.CA↗

arXiv2014

A hypergeometric basis for the Alpert multiresolution analysis

We construct an explicit orthonormal basis of piecewise ${}_{i+1}F_{i}$ hypergeometric polynomials for the Alpert multiresolution analysis. The Fourier transform of each basis function is written in terms of ${}_2F_3$ hypergeometric functions. Moreover, the entries in the matrix equation connecting the wavelets with the scaling functions are shown to be balanced ${}_4 F_3$ hypergeometric functions evaluated at $1$, which allows to compute them recursively via three-term recurrence relations. The above results lead to a variety of new interesting identities and orthogonality relations reminiscent to classical identities of higher-order hypergeometric functions and orthogonality relations of Wigner $6j$-symbols.

math.CA↗

arXiv2014

Wavelets centered on a knot sequence: theory, construction, and applications

We develop a general notion of orthogonal wavelets `centered' on an irregular knot sequence. We present two families of orthogonal wavelets that are continuous and piecewise polynomial. We develop efficient algorithms to implement these schemes and apply them to a data set extracted from an ocelot image. As another application, we construct continuous, piecewise quadratic, orthogonal wavelet bases on the quasi-crystal lattice consisting of the $τ$-integers where $τ$ is the golden ratio. The resulting spaces then generate a multiresolution analysis of $L^2(\mathbf{R})$ with scaling factor $τ$.

math.NA↗

arXiv2013

On Alpert Multiwavelets

The multiresolution analysis of Alpert is considered. Explicit formulas for the entries in the matrix coefficients of the refinement equation are given in terms of hypergeometric functions. These entries are shown to solve generalized eigenvalue equations as well as partial difference equations. The matrix coefficients in the wavelet equation are also considered and conditions are given to obtain a unique solution.

math.CA↗

arXiv2013

Polynomials with no zeros on a face of the bidisk

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement, namely no zeros on a face of the bidisk. Two different characterizations are given using a Hilbert space structure naturally associated to the trigonometric polynomial; one is in terms of a certain orthogonal decomposition the Hilbert space must possess called the "split-shift orthogonality condition" and another is an operator theoretic or matrix condition closely related to an earlier characterization due to the first two authors. This approach allows several refinements of the characterization and it also allows us to prove a sums of squares decomposition which at once generalizes the Cole-Wermer sums of squares result for two variable stable polynomials as well as a sums of squares result related to the Schur-Cohn method for counting the roots of a univariate polynomial in the unit disk.

math.CV↗

arXiv2012

Fejér-Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle

We give a complete characterization of the positive trigonometric polynomials Q(θ,ϕ) on the bi-circle, which can be factored as Q(θ,ϕ)=|p(e^{iθ},e^{iϕ})|^2 where p(z,w) is a polynomial nonzero for |z|=1 and |w|\leq 1. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight 1/(4π^2Q(θ,ϕ)) on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating the recurrence coefficients for the corresponding polynomials and vice versa. In particular, we characterize the Borel measures on the bi-circle for which the coefficients multiplying the reverse polynomials associated with the two operators: multiplication by z in lexicographical ordering and multiplication by w in reverse lexicographical ordering vanish after a particular point. This can be considered as a spectral type result analogous to the characterization of the Bernstein-Szegő measures on the unit circle.

math.CV↗

arXiv2011

Orthogonality relations for bivariate Bernstein-Szegő measures

The orthogonality properties of certain subspaces associated with bivariate Bernstein-Szegő measures are considered. It is shown that these spaces satisfy more orthogonality relations than expected from the relations that define them. The results are used to prove a Christoffel-Darboux like formula for these measures.

math.CV↗

arXiv2011

Parameters associated with bivariate Bernstein-Szego measures on the bi-circle

We consider measures supported on the bi-circle and review the recurrence relations satisfied by the orthogonal polynomials associated with these measures constructed using the lexicographical or reverse lexicographical ordering. New relations are derived among these recurrence coefficients. We extend the results of [8] on a parameterization for Bernstein-Szego measures supported on the bi-circle.

math.CA↗

arXiv2010

On Baxter's difference systems

We study the asymptotics of solutions of a difference system introduced by Baxter by using the general method for the asymptotic representation of such solutions due to Benzaid and Lutz. Some results of Tauberian type are obtained in the case when the spectral parameter belongs to the unit circle.

math.CA↗

arXiv2010

On the Markov sequence problem for Jacobi polynomials

We give a simple and entirely elementary proof of Gasper's theorem on the Markov sequence problem for Jacobi polynomials. It is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by Marc Kac. In the process, we obtain some new integral formulas for ratios of Jacobi polynomials that generalize Gasper's product formula and a well known formula of Koornwinder.

math.CA↗