On the oscillation of differential transforms of eigenfunction expansions
Authors:
C. L. Prather and J. K. Shaw
Journal:
Trans. Amer. Math. Soc. 280 (1983), 187206
MSC:
Primary 42C15; Secondary 30B50, 34B05
MathSciNet review:
712255
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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 SturmLiouville 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.
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 M. Abramowitz and I. Stegun (eds.), Hardbook 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. MR29 #4914. MR 0167642 (29:4914)
 [2]
 G. Birkhoff and G.C. Rota, Ordinary differential equations, Wiley, New York, 1969.
 [3]
 R. P. Boas and C. Prather, Final sets for operators on finite Fourier transforms, Houston Math. J. 5 (1979), 2936. MR 533636 (81b:42045)
 [4]
 E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGrawHill, New York, 1955. MR 0069338 (16:1022b)
 [5]
 E. Hille, On the oscillation of differential transform. II, Trans. Amer. Math. Soc. 52 (1942), 463497. MR 0007171 (4:97c)
 [6]
 , Lectures on ordinary differential equations, AddisonWesley, Reading, Mass., 1969. MR 0249698 (40:2939)
 [7]
 , Differential equations in the complex domain, Wiley, New York, 1976.
 [8]
 V. E. Kacnel'son, On the oscillation of derivatives of almost periodic functions, Teor. Funkciĭ Funkcional. Anal. i Priložen 2 (1966), 4254. (Russian) MR 0201916 (34:1794)
 [9]
 G. Pólya, On the zeros of the derivatives of a function and its analytic character, Bull. Amer. Math. Soc. 49 (1943), 178191. MR 0007781 (4:192d)
 [10]
 G. Pólya and N. Wiener, On the oscillations of the derivatives of a periodic function, Trans. Amer. Math. Soc. 52 (1942), 245256. MR 0007169 (4:97a)
 [11]
 C. Prather, On some old and new theorems on final sets, Houston Math. J. 7 (1981), 407430. MR 640983 (83b:30027)
 [12]
 , Final sets for operators on classes of entire functions representable as a Fourier integral, J. Math. Anal. Appl. 82 (1981), 200220. MR 626749 (83h:30023)
 [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 SturmLiouville expansions, Trans. Amer. Math. Soc. 257 (1980), 483505. MR 552270 (81b:34020)
 [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), 111119. MR 646377 (83h:30008)
 [17]
 , A Pólya "shire" theorem for functions with algebraic singularities, Internat. J. Math. Math. Sci. 5 (1982), 691706. MR 679411 (84a:30011)
 [18]
 G. Szegö, On the oscillation of differential transforms. I, Trans. Amer. Math. Soc. 52 (1942), 450462. MR 0007170 (4:97b)
 [19]
 , On the oscillation of differential transform. IV, Trans. Amer. Math. Soc. 53 (1943), 463468. MR 0008100 (4:244d)
 [20]
 E. C. Titchmarsh, Eigenfunction expansions associated with second order differential equations. Part I, Clarendon Press, Oxford, 1962. MR 0176151 (31:426)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198307122559
PII:
S 00029947(1983)07122559
Keywords:
Eigenfunction expansion,
iterates of operators,
sign changes
Article copyright:
© Copyright 1983 American Mathematical Society
