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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the oscillatory behavior of singular Sturm-Liouville expansions
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by J. K. Shaw PDF
Trans. Amer. Math. Soc. 257 (1980), 483-505 Request permission

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
  • © Copyright 1980 American Mathematical Society
  • 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