Results in Mathematics

, Volume 47, Issue 1–2, pp 93–114

# Eigenfunctions of Two-Scale Difference Equations with Dilation Parameter and Infinite Products

Article

## Abstract

This paper deals with two-scale difference equations having an arbitrary dilation parameter and a formal power series as symbol. We investigate the equation concerning the existence of nonzero compactly supported distributional solutions. In order to include also continuous solutions it is advantageous to consider the two-scale difference equation as eigenvalue problem where the solutions are either compactly supported or integrals of compactly supported distributions. Such solutions are called eigenfunctions. As main result we determine the necessary and sufficient condition for the existence of eigenfunctions that the symbol must be a rational function with a special structure depending on the dilation parameter. We show that the eigenfunctions can be expressed by means of a finite sum of shifted eigenfunctions belonging to the case with a polynomial symbol (characteristic polynomial), which is well investigated.

## Key words

Two-scale difference equations dilation parameter distributional solutions eigenfunctions rational symbols infinite products with rational factors

## AMS subject classification

39A13 39B32 46F10 44A10 26C15 12D05

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