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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On the oscillation of differential transforms of eigenfunction expansions


Authors: C. L. Prather and J. K. Shaw
Journal: Trans. Amer. Math. Soc. 280 (1983), 187-206
MSC: Primary 42C15; Secondary 30B50, 34B05
MathSciNet review: 712255
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Abstract: This paper continues the study of Pólya and Wiener, Hille and Szegö into the connections between the oscillation of derivatives of a real function and its analytic character. In the present paper, a Sturm-Liouville operator $ L$ is applied successively to an infinitely differentiable function which admits a certain eigenfunction expansion. The eigenfunction expansion is assumed to be "conservative", in the sense of Hille. Several theorems are given which link the frequency of oscillation of $ ({L^k}f)(x)$ to the size of the coefficients of $ f(x)$, and thus to its analytic character.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0712255-9
PII: S 0002-9947(1983)0712255-9
Keywords: Eigenfunction expansion, iterates of operators, sign changes
Article copyright: © Copyright 1983 American Mathematical Society