Generalised differences and a class of multiplier operators in Fourier analysis

TL;DR

This paper characterizes the range of certain second order differential operators on Sobolev spaces over the circle using generalized differences linked to the zeros of their multipliers, extending to higher orders and applications.

Contribution

It introduces a novel description of operator ranges via generalized differences involving zeros of multipliers, extending previous differentiation operator results.

Findings

Range characterized by five generalized second order differences

Extension to higher order operators and differences

Applications to automatic continuity of linear forms

Abstract

The ranges of a certain type of second order differential operator, on a Sobolev subspace of the Lebesgue space $L^2$ of the circle group, can be characterised by the vanishing of the Fourier coefficients at (generally) two integers that are the zeros of the multiplier of the operator. It is proved here that the range of any such operator may be alternatively described as comprising those functions in $L^2$ that are the sum of five generalised second order differences, each such difference involving the zeros of the multiplier. In fact, higher order operators and differences are considered. There are applications to automatic continuity of linear forms on $L^2$. This work is related to earlier work of G. Meisters and W. Schmidt who derived, in effect, a description of the range of the ordinary differentiation operator D (whose multiplier vanishes at 0) in terms of first order differences.