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On the divergence in rth mean of a class of eigenfunction expansions

Published online by Cambridge University Press:  14 November 2011

C. G. C. Pitts
Affiliation:
School of Mathematics and Physics, University of East Anglia, Norwich NR4 7TJ

Synopsis

We consider the expansion of a function in Lr (the class of measurable functions whose rth powers are Lebesgue integrable over some interval) in terms of the eigenfunctions arising from a singular Sturm-Liouville problem defined over an infinite or semi-infinite interval. We show that if l ≦ r ≦ inline1 or if r ≧ 4 there exists f in Lr whose eigenfunction expansion is divergent in the rth mean sense, and that the terms of the series form an unbounded sequence in Lr The result extends some work of Askey and Wainger concerning the Hermite series expansions of functions in Lr(–∞, ∞).

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1979

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