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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A singular Sturm-Liouville 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 .

**[1]**M. Abramowitz and I. A. Stegun (Eds.),*Handbook of mathematical functions with formulas, graphs and mathematical tables*, Nat. Bur. Standards Appl. Math. Ser., no. 55, Superintendent of Documents, U. S. Government Printing Office, Washington, D. C., 1964. MR**29**#4914. MR**0167642 (29:4914)****[2]**J. D. Buckholtz and J. K. Shaw,*Generalized completely convex functions and Sturm-Liouville operators*, SIAM J. Math. Anal.**6**(1975), 812-828. MR**0417484 (54:5534)****[3]**E. A. Coddington and N. Levinson,*Theory of ordinary differential equations*, McGraw-Hill, New York, 1955. MR**0069338 (16:1022b)****[4]**E. Hille,*On the oscillation of differential transforms*. II, Trans. Amer. Math. Soc.**52**(1942), 463-497. MR**0007171 (4:97c)****[5]**-,*Lectures on ordinary differential equations*, Addison-Wesley, Reading, Mass., 1969. MR**0249698 (40:2939)****[6]**G. Polya and N. Wiener,*On the oscillation of the derivatives of a periodic function*, Trans. Amer. Math. Soc.**52**(1942), 249-256. MR**0007169 (4:97a)****[7]**J. K. Shaw,*Analytic properties of generalized completely convex functions*, SIAM J. Math. Anal.**8**(1977), 271-279. MR**0432970 (55:5949)****[8]**J. K. Shaw and W. R. Winfrey,*Positivity properties of linear differential operators*, J. Math. Anal. Appl.**65**(1978), 184-200. MR**501747 (80b:34008)****[9]**G. Szegö,*On the oscillation of differential transforms*. I, Trans. Amer. Math. Soc.**52**(1942), 450-462. MR**0007170 (4:97b)****[10]**E. C. Titchmarsh,*Eigenfunction expansions associated with second-order differential equations*, Vol. I, Clarendon Press, Oxford, 1962. MR**0176151 (31:426)****[11]**D. V. Widder,*Functions whose even derivatives have a prescribed sign*, Proc. Nat. Acad. Sci. U.S.A.**26**(1940), 657-659. MR**0003437 (2:219f)****[12]**-,*Completely convex functions and Lidstone series*, Trans. Amer. Math. Soc.**51**(1942), 387-398. MR**0006356 (3:293b)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
34B25,
42C15

Retrieve articles in all journals with MSC: 34B25, 42C15

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