  The problem of applying Nash-Moser Newton methods to obtain periodic
solutions of the compressible Euler equations has led 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.
