Analog for the Wiener Lemma for Wolff-Denjoy Series

TL;DR

This paper proves an analogue of the Wiener lemma for Wolff-Denjoy series with real poles, showing that subtracting the linear part of 1/f yields a fractional part that also expands into Wolff-Denjoy series, with applications to operator theory.

Contribution

It establishes a Wiener lemma analogue for Wolff-Denjoy series, extending classical Fourier analysis results to this class of series with real poles.

Findings

The fractional part of 1/f also expands into Wolff-Denjoy series with negative coefficients.

The poles of the fractional part are real, mirroring the original series.

Applications to operator theory demonstrate the practical relevance of the result.

Abstract

Let a function f with real poles be expanded in a Wolff-Denjoy series with positive coefficients. The main result of the note states that if we subtract its linear part from the function 1/f, then the remaining fractional part of this function will also expand into Wolff-Denjoy series (its poles are also real, and the coefficients of the series are negative). In other words, for Wolff-Denjoy series of the indicated form, an analogue of the well-known Wiener lemma in the theory of Fourier series is true up to a linear term. Applications of the result to operator theory are given.