A multi-term basis criterion for families of dilated periodic functions

TL;DR

This paper introduces a concrete method to determine when dilated periodic functions form a Riesz basis in L^2(0,1), improving previous criteria by focusing on zero-pole localization of an analytic multiplier.

Contribution

It develops a practical approach based on a framework by Hedenmalm, Lindqvist, and Seip, for verifying basis properties through analytical and numerical methods, and applies it to specific function families.

Findings

Established a threshold for p-sine functions to form a Riesz basis.

Improved criteria for basis verification over previous estimates.

Demonstrated the method's effectiveness on profiles near non-basis families.

Abstract

In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in $L^2(0,1)$. This method relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients. Our focus is on the concrete verification of the hypotheses by means of analytical or accurate numerical approximations. We then examine the basis question for profiles in a neighbourhood of a non-basis family generated by periodic jump functions. For one of these profiles, the $p$-sine functions, we determine a threshold for positive answer to the basis question which improves upon those found recently.