Bernstein and Bernstein-Like Inequalities for Modulation Spaces

Sa'Ud Al-Sa'Di,Salti Samarah

doi:10.54938/ijemdm.2022.01.1.3

Abstract

It is shown that the modulation spaces Mwp can be characterized by the approximation behavior of their elements using Local Fourier bases. In analogy to the Local Fourier bases, we show that the modulation spaces can also be characterized by the approximation behavior of their elements using Gabor frames. We derive direct and inverse approximation theorems that describe the best approximation by linear combinations of N terms of a given function using its modulates and translates.

Highlights

One of the central problems of approximation theory is to characterize the set of functions which have a prescribed order of approximation by a given method of approximation, e.g., to characterize the functions which have approximation order O(N−α ) for some fixed α > 0
In this paper we consider the method of nonlinear approximation in particular spaces called the modulation spaces
Nonlinear approximation is utilized in many numerical algoritheorems, it occurs in several applications

Summary

Introduction

One of the central problems of approximation theory is to characterize the set of functions which have a prescribed order of approximation by a given method of approximation, e.g., to characterize the functions which have approximation order O(N−α ) for some fixed α > 0. Let ΣN(D) be the set in which the approximation is seeked, which denotes the collection of all functions in X which can be written as a linear combination of at most N elements of D, i.e., ΣN(D) = {s ∈ X; s = ∑ ckgk , gk ∈ D, ck ∈ C, card(F) ≤ N} k∈F (1.2) From this definition we note that the sum of two elements from ΣN is generally not in ΣN, and needs 2N terms in its representation by the gk’s, so it is rather to be in the larger set Σ2N, which means that the space ΣN is not linear, for this reason best N-term approximation is often called nonlinear approximation. We are interested in characterization the approximation space Ap(D) in the whole range of the parameters α, p and a given dictionary D

Gabor Frames and Modulation Spaces

Characterization of Modulation Spaces Mp