On subordination of linear operators on the Lebesgue spaces over $\mathbb R^d$

TL;DR

This paper introduces a general method using Fourier transforms of measures to compare linear operators on Lebesgue spaces, focusing on approximation methods and differential operators.

Contribution

It proposes a novel approach for operator comparison based on Fourier transforms, applicable to summability methods and differential operators.

Findings

Comparison of approximation properties of Fourier summability methods

Analysis of differential operators with constant coefficients

A unified framework for operator comparison

Abstract

In this note a general approach is suggested for comparison of operators. This is done by means of the Fourier transform of a measure. This approach is applied to comparison of approximation properties of various summability methods of the Fourier integrals (I) and to differential operators with constant coefficients (II).