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Zeitschrift für Analysis und ihre Anwendungen

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Volume 38, Issue 1, 2019, pp. 107–124
DOI: 10.4171/ZAA/1630

Published online: 2019-01-07

A Multi-Term Basis Criterion for Families of Dilated Periodic Functions

Lyonell Boulton[1] and Houry Melkonian[2]

(1) Heriot-Watt University, Edinburgh, UK
(2) University of Exeter, UK

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.

Keywords: Bases of dilated periodic functions, $p$-trigonometric functions, full equivalence to the Fourier basis

Boulton Lyonell, Melkonian Houry: A Multi-Term Basis Criterion for Families of Dilated Periodic Functions. Z. Anal. Anwend. 38 (2019), 107-124. doi: 10.4171/ZAA/1630