Sobolev spaces of fractional order, Lipschitz spaces, readapted modulation spaces and their interrelations; applications

The purpose of this investigation is to extend basic equations and inequalities which hold for functions ff in a Bernstein space Bσ2 to larger spaces by adding a remainder term which involves the distance of ff from Bσ2.First we present a modification of the classical modulation space M2,1(R)M2,1(R), the so-called readapted modulation space Ma2,1(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 Ma2,1(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’skiĭ’s inequality estimating the lp(Z)lp(Z) norm in terms of the Lp(R)Lp(R) norm.