A multi-term basis criterion for families of dilated periodic functions
Abstract
In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in . 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 -sine functions, we determine a threshold for positive answer to the basis question which improves upon those found recently.
Keywords
Cite
@article{arxiv.1708.08545,
title = {A multi-term basis criterion for families of dilated periodic functions},
author = {Lyonell Boulton and Houry Melkonian},
journal= {arXiv preprint arXiv:1708.08545},
year = {2017}
}
Comments
The introduction of the updated version now includes an example which is arbitrarily close to the jump function such that the corresponding dilations form a Riesz basis. Section 2 has been substantially changed to reflect the established multi-term criterion in terms of the multipliers instead of the corresponding operators. The paper consists of 26 pages and 4 figures