Translation beyond Delsarte

TL;DR

This paper introduces a general translation framework using solutions to boundary value problems, enabling new characterizations of function spaces and smoothness measures in analysis.

Contribution

It presents a novel approach to defining translations via PDE solutions, leading to new characterizations of compactness and smoothness in function spaces.

Findings

Kolmogorov-Riesz characterization of compact sets in L^p spaces

Pego-type characterizations of smoothness

Equivalence of modulus of smoothness and K-functional

Abstract

We introduce general translations as solutions to Cauchy or Dirichlet problems. This point of view allows us to handle the heat-diffusion semigroup as a translation. With the given examples Kolmogorov-Riesz characterization of compact sets in certain $L^p_\mu$ spaces are given. Pego-type characterizations are also derived. Finally for some examples the equivalence of the corresponding modulus of smoothness and K-functional is pointed out.