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24.—Mean-square Convergence of Non-harmonic Trigonometrical Series

Published online by Cambridge University Press:  14 February 2012

J. Cossar
J. Cossar
Affiliation:
Department of Mathematics, University of Edinburgh
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Synopsis

The series considered are of the form , where Σ | cn |2 is convergent and the real numbers λn (the exponents) are distinct. It is known that if the exponents are integers, the series is the Fourier series of a periodic function of locally integrable square (the Riesz-Fischer theorem); and more generally that if the exponents are not necessarily integers but are such that the difference between any pair exceeds a fixed positive number, the series is the Fourier series of a function of the Stepanov class, S2, of almost periodic functions.

We consider in this paper cases where the exponents are subject to less stringent conditions (depending on the coefficients cn). Some of the theorems included here are known but had been proved by other methods. A fuller account of the contents of the paper is given in Sections 1-5.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 75 , Issue 4 , 1976 , pp. 297 - 323
Copyright
Copyright © Royal Society of Edinburgh 1976

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