Methods and Applications of Analysis

Volume 27 (2020)

Number 1

Inversion of a non-uniform difference operator

Pages: 65 – 86



Blake Temple (Department of Mathematics, University of California at Davis)

Robin Young (Department of Mathematics and Statistics, University of Massachusetts, Amherst, Ma., U.S.A.)


The problem of applying Nash–Moser Newton methods to obtain periodic solutions of the compressible Euler equations has led the authors to identify the main obstacle, namely, how to invert operators which impose periodicity when they are based on non-uniform shift operators. Here we begin a theory for finding the inverses of such operators by proving that a scalar non-uniform difference operator does in fact have a bounded inverse on its range. We argue that this is the simplest example which demonstrates the need to use direct rather than Fourier methods to analyze inverses of linear operators involving nonuniform shifts.


periodic solutions, shock waves, Nash–Moser, nonlinear shift operator

2010 Mathematics Subject Classification


Received 19 June 2019

Accepted 11 February 2020

Published 12 August 2020