Generalised differences and multiplier operators in $L^2({\mathbb R})$

TL;DR

This paper characterizes functions in $L^2(\mathbb{R})$ based on their Fourier transform behavior near specific points, using generalized differences that approximate differential operators, with implications for Sobolev spaces and invariant forms.

Contribution

It introduces a new characterization of $L^2(\mathbb{R})$ functions via generalized differences linked to differential operators, extending previous circle group results.

Findings

Functions with Fourier transforms vanishing near certain points are sums of generalized differences.

Range of differential operators on Sobolev spaces corresponds to functions expressible as finite sums of generalized differences.

Connections established between these differences, differential operators, and invariant forms on $L^2$ spaces.

Abstract

Given two real numbers, the $L^2$ functions whose Fourier transforms vanish with a certain rapidity near the given numbers are characterised as those that are expressible as the sum of a certain number of generalised finite differences that is independent of the function. These generalised differences can be regarded as approximating the appropriate powers of first order ordinary differential operators. The upshot of this is that for operators in a certain class of ordinary differential operators that have polynomial multipliers, their ranges on the Sobolev spaces corresponding to the operators are those functions expressible as a finite sum of corresponding generalised differences, so that the latter form a weighted $L^2$ space under the Fourier transform. There is a connection with the continuity properties of invariant forms on $L^2$ spaces. The results presented here complement results previously obtained for the $L^2$ space of the circle group.