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On the Oscillation of Differential Transforms of Eigenfunction Expansions

C. L. Prather and J. K. Shaw
Transactions of the American Mathematical Society
Vol. 280, No. 1 (Nov., 1983), pp. 187-206
DOI: 10.2307/1999608
Stable URL: http://www.jstor.org/stable/1999608
Page Count: 20
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On the Oscillation of Differential Transforms of Eigenfunction Expansions
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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 $(L^kf)(x)$ to the size of the coefficients of $f(x)$, and thus to its analytic character.

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