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On the oscillatory behavior of singular Sturm-Liouville expansions


Author: J. K. Shaw
Journal: Trans. Amer. Math. Soc. 257 (1980), 483-505
MSC: Primary 34B25; Secondary 42C15
DOI: https://doi.org/10.1090/S0002-9947-1980-0552270-9
MathSciNet review: 552270
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Abstract: A singular Sturm-Liouville operator $ Ly\, =\, - (Py')'\, +\, Qy$, defined on an interval $ [0,b^{\ast})$ of regular points, but singular at $ b^{\ast}$, is considered. Examples are the Airy equation on $ [0,\infty )$ and the Legendre equation on $ [0,1)$. A mode of oscillation of the successive iterates $ f(t)$, $ (Lf)(t)$, $ ({L^2}f)(t),\, \ldots $ of a smooth function f is assumed, and the resulting influence on f is studied. The nature of the mode is that for a fixed integer $ N\, \geqslant\, 0$, each iterate $ ({L^k}f)(t)$ shall have on $ (0,b^{\ast})$ exactly N sign changes which are stable, in a certain sense, as k varies. There is quoted from the literature the main characterization of such functions f which additionally satisfy strong homogeneous endpoint conditions at 0 and $ b^{\ast}$. An extended characterization is obtained by weakening the conditions of f at 0 and $ b^{\ast}$. The homogeneous endpoint conditions are replaced by a summability condition on the values, or limits of values, of f at 0 and $ b^{\ast}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0552270-9
Keywords: Eigenfunction expansion, iterates of operators, sign changes
Article copyright: © Copyright 1980 American Mathematical Society

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