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Isaac Z. Pesenson

Publications and source records attributed to Isaac Z. Pesenson.

At least 19 recordsLinked to original sources
arXiv2024

To Multidimensional Mellin Analysis: Besov spaces, $K$-functor, approximations, frames

In the setting of the multidimensional Mellin analysis we introduce moduli of continuity and use them to define Besov-Mellin spaces. We prove that Besov-Mellin spaces are the interpolation spaces (in the sense of J.Peetre) between two Sobolev-Mellin spaces. We also introduce Bernstein-Mellin spaces and prove corresponding direct and inverse approximation theorems. In the Hilbert case we discuss Laplace-Mellin operaor and define relevant Paley-Wiener-Mellin spaces. Also in the Hilbert case we describe Besov-Mellin spaces in terms of Hilbert frames.

math.FA↗
arXiv2023

Jackson theorem and modulus of continuity in Hilbert spaces and on homogeneous manifolds

We consider a Hilbert space ${\bf H}$ equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. The conditions allow to define a family of moduli of continuity $Ω^{r}(s,f),\>r\in \mathbb{N}, s>0,$ of vectors in ${\bf H}$ and a family of Paley-Wiener subspaces $PW_σ$ parametrized by bandwidth $σ>0$. These subspaces are explored to introduce notion of the best approximation $\mathcal{E}(σ, f)$ of a general vector in ${\bf H}$ by Paley-Wiener vectors of a certain bandwidth $σ>0$. The main objective of the paper is to prove the so-called Jackson-type estimate $\mathcal{E}(σ, f)\leq C\left( Ω^{r}(σ^{-1},f)+σ^{-r}\|f\|\right)$ for $σ>1$. It was shown in our previous publications that our assumptions are satisfied for a strongly continuous unitary representation of a Lie group $G$ in a Hilbert space ${\bf H}$. This way we obtain the Jackson-type estimates on homogeneous manifolds.

math.FA↗
arXiv2023

Sobolev, Besov and Paley-Wiener vectors in Banach and Hilbert spaces

We consider Banach spaces equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. Using these semigroups we introduce an analog of a modulus of continuity and define analogs of Besov norms. A generalization of a classical interpolation theorem is proven in which the role of Sobolev spaces is played by subspaces defined in terms of infinitesimal operators of these semigroups. We show that our assumptions about a given set of semigroups are satisfied in the case of a strongly continuous bounded representation of a Lie group. In the case of a unitary representation in a Hilbert space we consider an analog of the Laplace operator and use it to define Paley-Wiener vectors. It allows us to develop a generalization of the Shannon-type sampling in Paley-Wiener subspaces and to construct Paley-Wiener nearly Parseval frames in the entire Hilbert space. It is shown that Besov spaces defined previously in terms of the modulus of continuity can be described in terms of approximation by Paley-Wiener vectors and also in terms of the frame coefficients. Throughout the paper we extensively use theory of interpolation and approximation spaces. The paper ends with applications of our results to function spaces on homogeneous manifolds.

math.FA↗
arXiv2019

A Weak Weyl's Law on compact metric measure spaces

The well known Weyl's Law (Weyl's asymptotic formula) gives an approximation to the number $\mathcal{N}_ω$ of eigenvalues (counted with multiplicities) on a large interval $[0,\>ω]$ of the Laplace-Beltrami operator on a compact Riemannian manifold ${\bf M}$. In this paper we prove a kind of a weak version of the Weyl's law on certain compact metric measure spaces ${\bf X}$ which are equipped with a self-adjoint non-negative operator $\mathcal{L}$ acting in $L_{2}({\bf X})$. Roughly speaking, we show that if a certain Poincaré inequality holds then $\mathcal{N}_ω$ is controlled by the cardinality of an appropriate cover $\mathcal{B}_{ω^{-1/2}}=\{B(x_{j},ω^{-1/2})\},\>\>\>x_{j}\in {\bf X},$ of ${\bf X}$ by balls of radius $ω^{-1/2}$. Moreover, an opposite inequality holds if the heat kernel that corresponds to $\mathcal{L}$ satisfies short time Gaussian estimates. It is known that in the case of the so-called strongly local regular with a complete intrinsic metric Dirichlet spaces the Poincaré inequality holds iff the corresponding heat kernel satisfies short time Gaussian estimates. Thus for such spaces one obtains that $\mathcal{N}_ω$ is essentially equivalent to the cardinality of a cover $\mathcal{B}_{ω^{-1/2}}$.

math.FA↗
arXiv2019

Sampling by averages and average splines on Dirichlet spaces and on combinatorial graphs

In the framework of a strictly local regular Dirichlet space ${\bf X}$ we introduce the subspaces $PW_ω,\>\>ω>0,$ of Paley-Wiener functions of bandwidth $ω$. It is shown that every function in $PW_ω,\>\>ω>0,$ is uniquely determined by its average values over a family of balls $B(x_{j}, ρ),\>x_{j}\in {\bf X},$ which form an admissible cover of ${\bf X}$ and whose radii are comparable to $ω^{-1/2}$. The entire development heavily depends on some local and global Poincaré-type inequalities. In the second part of the paper we realize the same idea in the setting of a weighted combinatorial finite or infinite countable graph $G$. We have to treat the case of graphs separately since the Poincaré inequalities we are using on them are somewhat different from the Poincaré inequalities in the first part.

math.FA↗
arXiv2019

Weighted sampling and weighted interpolation on combinatorial graphs

For Paley-Wiener functions on weighted combinatorial finite or infinite graphs we develop a weighted sampling theory in which samples are defined as inner products with weight functions (measuring devices). Three reconstruction methods are suggested. The first two of them are using language of dual Hilbert frames and the so-called frame algorithm respectively. The third one is using the so-called weighted variational interpolating splines which are constructed in the setting of combinatorial graphs. This development requires a new set of Poincaré-type inequalities which we prove for functions on combinatorial graphs.

math.FA↗
arXiv2019

Cubature formulas on combinatorial graphs

The goal of the paper is to establish cubature formulas on combinatorial graphs. Two types of cubature formulas are developed. Cubature formulas of the first type are exact on spaces of variational splines on graphs. Since badlimited functions can be obtained as limits of variational splines we obtain cubature formulas which are "essentially" exact on spaces of bandlimited functions. Cubature formulas of the second type are exact on spaces of bandlimited functions. Accuracy of cubature formulas is given in terms of smoothness which is measured by means of combinatorial Laplace operator. The results have potential applications to problems that arise in data mining.

math.FA↗
arXiv2017

Shannon sampling and Weak Weyl's Law on compact Riemannian manifolds

The well known Weyl's asymptotic formula gives an approximation to the number $\mathcal{N}_ω$ of eigenvalues (counted with multiplicities) on an interval $[0,\>ω]$ of the Laplace-Beltrami operator on a compact Riemannian manifold ${\bf M}$. In this paper we approach this question from the point of view of Shannon-type sampling on compact Riemannian manifolds. Namely, we give a direct proof that $\mathcal{N}_ω$ is comparable to cardinality of certain sampling sets for the subspace of $ω$-bandlimited functions on ${\bf M}$.

math.FA↗
arXiv2015

Geometric Space-Frequency Analysis on Manifolds

This paper gives a survey of methods for the construction of space-frequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general context, the notion of frequency is defined using the spectrum of a distinguished differential operator on the manifold, typically the Laplace-Beltrami operator. Our exposition starts with the case of the real line, which serves as motivation and blueprint for the material in the subsequent sections. After the discussion of the real line, our presentation starts out in the most abstract setting proving rather general sampling-type results for appropriately defined Paley-Wiener vectors in Hilbert spaces. These results allow a handy construction of Paley-Wiener frames in $L_2(\mfd{M})$, for a Riemann manifold of bounded geometry, essentially by taking a partition of unity in frequency domain. The discretization of the associated integral kernels then gives rise to frames consisting of smooth functions in $L_2(\mfd{M})$, with fast decay in space and frequency. These frames are used to introduce new norms in corresponding Besov spaces on $\mfd{M}$. For compact Riemannian manifolds the theory extends to $L_p$ and Besov spaces. Moreover, for compact homogeneous manifolds, one obtains the so-called product property for eigenfunctions of certain operators and proves a cubature formulae with positive coefficients which allow to construct Parseval frames that characterize Besov spaces in terms of coefficient decay. Throughout the paper, the general theory is exemplified with the help of various concrete and relevant examples, such as the unit sphere and the Poincaré half plane.

math.FA↗
arXiv2015

Estimates of Kolmogorov, Gelfand and linear $n$- widths on Compact Riemannian Manifolds

We determine lower and exact estimates of Kolmogorov, Gelfand and linear $n$-widths of unit balls in Sobolev norms in $L_{p}$-spaces on compact Riemannian manifolds. As it was shown by us previously these lower estimates are exact asymptotically in the case of compact homogeneous manifolds. The proofs rely on two-sides estimates for the near-diagonal localization of kernels of functions of elliptic operators.

math.CA↗
arXiv2015

Sampling solutions of Schrödinger equations on combinatorial graphs

We consider functions on a graph $G$ whose evolution in time $-\infty 0$ such that solutions to a Cauchy problem with initial data in $PW_ω(G)$ are completely determined by their samples on $S\times \{kπ/ω\},$ where $k\in \mathbf{N}$. It is shown that in the case of a bipartite graph our results are sharp.

math.SP↗
arXiv2015

Sampling, splines and frames on compact 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 the Shannon-type sampling, 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↗
arXiv2014

A Simple Proposal for Radial 3D Needlets

We present here a simple construction of a wavelet system for the three-dimensional ball, which we label \emph{Radial 3D Needlets}. The construction envisages a data collection environment where an observer located at the centre of the ball is surrounded by concentric spheres with the same pixelization at different radial distances, for any given resolution. The system is then obtained by weighting the projector operator built on the corresponding set of eigenfunctions, and performing a discretization step which turns out to be computationally very convenient. The resulting wavelets can be shown to have very good localization properties in the real and harmonic domain; their implementation is computationally very convenient, and they allow for exact reconstruction as they form a tight frame systems. Our theoretical results are supported by an extensive numerical analysis.

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