On the oscillation of differential transforms of eigenfunction expansions

Authors:
C. L. Prather and J. K. Shaw

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

MSC:
Primary 42C15; Secondary 30B50, 34B05

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

MathSciNet review:
712255

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 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 to the size of the coefficients of , and thus to its analytic character.

**[1]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[2]**G. Birkhoff and G.-C. Rota,*Ordinary differential equations*, Wiley, New York, 1969.**[3]**R. P. Boas Jr. and Carl L. Prather,*Final sets for operators on finite Fourier transforms*, Houston J. Math.**5**(1979), no. 1, 29–36. MR**533636****[4]**Earl A. Coddington and Norman Levinson,*Theory of ordinary differential equations*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR**0069338****[5]**Einar Hille,*On the oscillation of differential transforms. II. Characteristic series of boundary value problems*, Trans. Amer. Math. Soc.**52**(1942), 463–497. MR**0007171**, https://doi.org/10.1090/S0002-9947-1942-0007171-8**[6]**Einar Hille,*Lectures on ordinary differential equations*, Addison-Wesley Publ. Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR**0249698****[7]**-,*Differential equations in the complex domain*, Wiley, New York, 1976.**[8]**V. È. Kacnel′son,*Oscillation of the derivatives of almost periodic functions*, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp.**2**(1966), 42–54 (Russian). MR**0201916****[9]**G. Polya,*On the zeros of the derivatives of a function and its analytic character*, Bull. Amer. Math. Soc.**49**(1943), 178–191. MR**0007781**, https://doi.org/10.1090/S0002-9904-1943-07853-6**[10]**George Pólya and Norbert Wiener,*On the oscillation of the derivatives of a periodic function*, Trans. Amer. Math. Soc.**52**(1942), 249–256. MR**0007169**, https://doi.org/10.1090/S0002-9947-1942-0007169-X**[11]**C. L. Prather,*On some new and old theorems, on final sets*, Houston J. Math.**7**(1981), no. 3, 407–430. MR**640983****[12]**C. L. Prather,*Final sets for operators on classes of entire functions representable as a Fourier integral*, J. Math. Anal. Appl.**82**(1981), no. 1, 200–220. MR**626749**, https://doi.org/10.1016/0022-247X(81)90233-X**[13]**-,*The oscillation of derivatives. The Bernstein problem for Fourier integrals*, submitted.**[14]**-,*The oscillation of differential transforms. The Bernstein problem for Hermitian and Laguerre expansions*, preprint.**[15]**J. K. Shaw,*On the oscillatory behavior of singular Sturm-Liouville expansions*, Trans. Amer. Math. Soc.**257**(1980), no. 2, 483–505. MR**552270**, https://doi.org/10.1090/S0002-9947-1980-0552270-9**[16]**J. K. Shaw and C. L. Prather,*Zeros of successive derivatives of functions analytic in a neighborhood of a single pole*, Michigan Math. J.**29**(1982), no. 1, 111–119. MR**646377****[17]**J. K. Shaw and C. L. Prather,*A Pólya “shire” theorem for functions with algebraic singularities*, Internat. J. Math. Math. Sci.**5**(1982), no. 4, 691–706. MR**679411**, https://doi.org/10.1155/S0161171282000635**[18]**G. Szegö,*On the oscillation of differential transforms. I*, Trans. Amer. Math. Soc.**52**(1942), 450–462. MR**0007170**, https://doi.org/10.1090/S0002-9947-1942-0007170-6**[19]**G. Szegö,*On the oscillation of differential transforms. IV. Jacobi polynomials*, Trans. Amer. Math. Soc.**53**(1943), 463–468. MR**0008100**, https://doi.org/10.1090/S0002-9947-1943-0008100-4**[20]**E. C. Titchmarsh,*Eigenfunction expansions associated with second-order differential equations. Part I*, Second Edition, Clarendon Press, Oxford, 1962. MR**0176151**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
42C15,
30B50,
34B05

Retrieve articles in all journals with MSC: 42C15, 30B50, 34B05

Additional Information

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

Keywords:
Eigenfunction expansion,
iterates of operators,
sign changes

Article copyright:
© Copyright 1983
American Mathematical Society