On the oscillation of differential transforms of eigenfunction expansions

Prather, C. L.; Shaw, J. K.

doi:10.1090/S0002-9947-1983-0712255-9

On the oscillation of differential transforms of eigenfunction expansions
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by C. L. Prather and J. K. Shaw

Trans. Amer. Math. Soc. 280 (1983), 187-206

DOI: https://doi.org/10.1090/S0002-9947-1983-0712255-9

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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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