Sobolev Spaces of Fractional Order, Lipschitz Spaces, Readapted Modulation Spaces and Their Interrelations; Applications

TL;DR

This paper introduces a new readapted modulation space and explores its relationships with Lipschitz and fractional Sobolev spaces, extending classical inequalities and formulas with applications in sampling theory and functional analysis.

Contribution

It presents a novel readapted modulation space and establishes new inclusion relations with Lipschitz and fractional Sobolev spaces, extending classical inequalities and formulas.

Findings

Introduction of the readapted modulation space $M^{2,1}_a(R)$

Chains of inclusion relations between spaces

Extensions of classical inequalities and formulas

Abstract

The purpose of this investigation is to extend basic equations and inequalities which hold for functions $f$ in a Bernstein space $B_\sigma^2$ to larger spaces by adding a remainder term which involves the distance of $f$ from $B_\sigma^2$. First we present a modification of the classical modulation space $M^{2,1}(\mathbb{R})$, the so-called readapted modulation space $M^{2,1}_\text{a}(\mathbb{R})$. Our approach to the latter space and its role in functional analysis is novel. In fact, we establish several chains of inclusion relations between $M^{2,1}_\text{a}(\mathbb{R})$ and the more common Lipschitz and Sobolev spaces, including Sobolev spaces of fractional order. Next we introduce an appropriate metric for describing the distance of a function belonging to one of the latter spaces from a Bernstein space. It will be used for estimating remainders and studying rates of convergence. In the main part, we present the desired extensions. Our applications include the classical Whittaker-Kotel'nikov-Shannon sampling formula, the reproducing kernel formula, the Parseval decomposition formula, Bernstein's inequality for derivatives, and Nikol'ski\u{\i}'s inequality estimating the $l^p(\mathbb{Z})$ norm in terms of the $L^p(\mathbb{R})$ norm.