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Isaac Pesenson

Publications and source records attributed to Isaac Pesenson.

16 recordsLinked to original sources
arXiv2025

Notes on Bernstein spaces, sampling, Boas interpolation formulas and their extensions to Banach spaces

This paper is essentially a survey on several classical results of harmonic analysis and their recent extensions to Banach spaces. The first part of the paper is a summary of some important results in such topics as Bernstein spaces, Shannon-type sampling, Riesz and Boas interpolation formulas. The second part contains extensions of these ideas to Banach spaces equipped with one-parameter uniformly bounded group of operators of class $C_{0}$.

math.FA↗
arXiv2024

Besov and Paley-Wiener spaces, Moduli of continuity and Hardy-Steklov operators associated with the group $"ax+b"$

We introduce and describe relations between Sobolev, Besov and Paley-Wiener spaces associated with three representations of the Lie group of affine transformations of the line. These representations are left and right regular representations and a representation in a space of functions defined on the half-line. The Besov spaces are described as interpolation spaces between respective Sobolev spaces in terms of the Petree's real interpolation method and in terms of a relevant moduli of continuity. By using a Laplace operators associated with these representations a scales of relevant Paley-Wiener spaces are developed and a corresponding $L_{2}$-approximation theory is constructed in which our Besov spaces appear as approximation spaces. Another description of our Besov spaces is given in terms of a frequency-localized Hilbert frames. A Jackson-type inequalities are also proven.

math.FA↗
arXiv2023

Exact positive cubature formulas via generalized sampling on combinatorial graphs

We consider a disjoint cover (partition) of an undirected weighted finite graph $G$ by $|J|$ connected subgraphs (clusters) $\{S_{j}\}_{j\in J}$ and select a function $\zeta_{j}\geq 0$ on each of the clusters. For a given signal $f$ on $G$ the set of its weighted average values samples is defined via inner products $\{\langle \zeta_{j}, f\rangle\}_{j\in J}$. The goal of the paper is to establish exact quadrature formulas with positive weights which are exploring these samples generated by bandlimited functions.

math.FA↗
arXiv2014

$n$-widths and Approximation theory on Compact Riemannian Manifolds

We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on compact Riemannian manifolds. The proofs rely on estimates for the near-diagonal localization of the kernels of elliptic operators. We also summarize some of our previous results about approximations by eigenfunctions of elliptic operators on compact homogeneous manifolds.

math.FA↗
arXiv2014

Splines and Wavelets on Geophysically Relevant Manifolds

Analysis on the unit sphere $\mathbb{S}^{2}$ found many applications in seismology, weather prediction, astrophysics, signal analysis, crystallography, computer vision, computerized tomography, neuroscience, and statistics. In the last two decades, the importance of these and other applications triggered the development of various tools such as splines and wavelet bases suitable for the unit spheres $\mathbb{S}^{2}$, $\>\>\mathbb{S}^{3}$ and the rotation group $SO(3)$. Present paper is a summary of some of results of the author and his collaborators on generalized (average) variational splines and localized frames (wavelets) on compact Riemannian manifolds. The results are illustrated by applications to Radon-type transforms on $\mathbb{S}^{d}$ and $SO(3)$.

math.FA↗
arXiv2012

Kolmogorov and Linear Widths of Balls in Sobolev and Besov Norms on Compact Manifolds

We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on smooth compact Riemannian manifolds. For compact homogeneous manifolds, we establish estimates which are asymptotically exact, for the natural ranges of indices. The proofs heavily rely on our previous results such as: estimates for the near-diagonal localization of the kernels of elliptic operators, Plancherel-Polya inequalities on manifolds of bounded geometry, cubature formulas with positive coefficients and uniform estimates on Clebsch-Gordon coefficients on general compact homogeneous manifolds.

math.FA↗
arXiv2011

Sampling in paley-wiener spaces on combinatorial graphs

A notion of Paley-Wiener spaces is introduced on combinatorial graphs. It is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets. Such uniqueness sets are described in terms of Poincare-Wirtingertype inequalities. A reconstruction algorithm of Paley-Wiener functions from uniqueness sets which uses the idea of frames in Hilbert spaces is developed. Special consideration is given to n-dimensional lattice, homogeneous trees, and eigenvalue and eigenfunction problems on finite graphs.

math.SP↗
arXiv2011

Variational Splines and Paley--Wiener Spaces on Combinatorial Graphs

Notions of interpolating variational splines and Paley-Wiener spaces are introduced on a combinatorial graph G. Both of these definitions explore existence of a combinatorial Laplace operator onG. The existence and uniqueness of interpolating variational splines on a graph is shown. As an application of variational splines, the paper presents a reconstruction algorithm of Paley-Wiener functions on graphs from their uniqueness sets.

cs.IT↗
arXiv2011

Reconstruction of Paley-Wiener functions on the Heisenberg group

Let $M$ be a Riemmanian manifold with bounded geometry. We consider a generalization of Paley-Wiener functions and Lagrangian splines on $M$. An analog of the Paley-Wiener theorem is given. We also show that every Paley-Wiener function on a manifold is uniquely determined by its values on some discrete sets of points. The main result of the paper is a generalization of the Whittaker-Shannon formula for reconstruction of a Paley-Wiener function from its values on a discrete set. It is shown that every Paley- Wiener function on $M$ is a limit of some linear combinations of fundamental solutions of the powers of the Laplace-Beltrami operator. The result is new even in the one-dimentional case.

math.FA↗
arXiv2011

Irregular sampling and the Radon transform

In image reconstruction there are techniques that use analytical formulae for the Radon transform to recover an image from a continuum of data. In practice, however, one has only discrete data available. Thus one often resorts to sampling and interpolation methods. This article presents an approach to the inversion of the Radon transform that uses a discrete set of samples which need not be completely regular.

math.FA↗
arXiv2011

Deconvolution of band limited functions on non-compact symmetric spaces

It is shown that a band limited function on a non-compact symmetric space can be reconstructed in a stable way from some countable sets of values of its convolution with certain distributions of compact support. A reconstruction method in terms of frames is given which is a generalization of the classical result of Duffin-Schaeffer about exponential frames on intervals. The second reconstruction method is given in terms of polyharmonic average splines.

math.FA↗
arXiv2011

A Reconstruction Method for Band-Limited Signals on the Hyperbolic Plane

A notion of band limited functions is considered in the case of the hyperbolic plane in its Poincare upper half-plane $\mathbb{H}$ realization. The concept of band-limitedness is based on the existence of the Helgason-Fourier transform on $\mathbb{H}$. An iterative algorithm is presented, which allows to reconstruct band-limited functions from some countable sets of their values. It is shown that for sufficiently dense metric lattices a geometric rate of convergence can be guaranteed as long as the sampling density is high enough compared to the band-width of the sampled function.

math.FA↗
arXiv2011

Variational splines on Riemannian manifolds with applications to integral geometry

We extend the classical theory of variational interpolating splines to the case of compact Riemannian manifolds. Our consideration includes in particular such problems as interpolation of a function by its values on a discrete set of points and interpolation by values of integrals over a family of submanifolds. The existence and uniqueness of interpolating variational spline on a Riemannian manifold is proven. Optimal properties of such splines are shown. The explicit formulas of variational splines in terms of the eigen functions of Laplace-Beltrami operator are found. It is also shown that in the case of interpolation on discrete sets of points variational splines converge to a function in $C^{k}$ norms on manifolds. Applications of these results to the hemispherical and Radon transforms on the unit sphere are given.

math.FA↗
arXiv2011

Frames for spaces of Paley-Wiener functions on Riemannian manifolds

It is shown that Paley-Wiener functions on Riemannian manifolds of bounded geometry can be reconstructed in a stable way from some countable sets of their inner products with certain distributions of compact support. A reconstruction method in terms of frames is given which is a generalization of the classical result of Duffin-Schaeffer about exponential frames on intervals. All results are specified in the case of the two-dimensional hyperbolic space in its Poincare upper half-plane realization.

math.FA↗
arXiv2011

A Discrete Helgason-Fourier transform for Sobolev and Besov functions on noncompact symmetric spaces

Let $f$ be a Paley-Wiener function in the space $L_{2}(X)$, where $X$ is a symmetric space of noncompact type. It is shown that by using the values of $f$ on a sufficiently dense and separated set of points of $X$ one can give an exact formula for the Helgason-Fourier transform of $f$. In order to find a discrete approximation to the Helgason-Fourier transform of a function from a Besov space on $X$ we develop an approximation theory by Paley-Wiener functions in $L_{2}(X)$.

math.FA↗