A note on certain convolution operators

TL;DR

This paper investigates a specific class of convolution operators on L_p spaces over the one-dimensional torus, demonstrating their invertibility properties when adjusted by the identity on mean-zero functions.

Contribution

It introduces a new class of convolution operators and establishes their invertibility characteristics on mean-zero subspaces of L_p spaces.

Findings

Identity minus the operator is invertible on mean-zero functions

Convolution operators are well-behaved on L_p spaces

Provides theoretical insights into operator invertibility

Abstract

In this note we consider a certain class of convolution operators acting on the L_p spaces of the one dimensional torus. We prove that the identity minus such an operator is nicely invertible on the subspace of functions with mean zero.