On the oscillatory behavior of singular SturmLiouville expansions
Author:
J. K. Shaw
Journal:
Trans. Amer. Math. Soc. 257 (1980), 483505
MSC:
Primary 34B25; Secondary 42C15
MathSciNet review:
552270
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Abstract: A singular SturmLiouville operator , defined on an interval of regular points, but singular at , is considered. Examples are the Airy equation on and the Legendre equation on . A mode of oscillation of the successive iterates , , 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 , each iterate shall have on 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 . An extended characterization is obtained by weakening the conditions of f at 0 and . The homogeneous endpoint conditions are replaced by a summability condition on the values, or limits of values, of f at 0 and .
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 , Completely convex functions and Lidstone series, Trans. Amer. Math. Soc. 51 (1942), 387398. MR 0006356 (3:293b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198005522709
PII:
S 00029947(1980)05522709
Keywords:
Eigenfunction expansion,
iterates of operators,
sign changes
Article copyright:
© Copyright 1980
American Mathematical Society
