On the absolute convergence of lacunary Fourier series
Author:
J. R. Patadia
Journal:
Proc. Amer. Math. Soc. 71 (1978), 1925
MSC:
Primary 42A44
MathSciNet review:
0493138
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Abstract: Let be periodic. Noble [6] posed the following problem: if the fulfillment of some property of a function f on the whole interval implies certain conclusions concerning the Fourier series of f, then what lacunae in guarantees the same conclusions when the property is fulfilled only locally? Applying the more powerful methods of approach to this kind of problems, originally developed by Paley and Wiener [7], the absolute convergence of a certain lacunary Fourier series is studied when the function f satisfies some hypothesis in terms of either the modulus of continuity or the modulus of smoothness of order l considered only at a fixed point of . The results obtained here are a kind of generalization of the results due to Patadia [8].
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A. Zygmund, Trigonometrical series, Warsaw, 1935.
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 N. K. Bary, A treatise on trigonometric series, Vol. II, Pergamon Press, New York, 1964.
 [2]
 Jiaarng Chao, On Fourier series with gaps, Proc. Japan Acad. 42 (1966), 308312. MR 0203338 (34:3191)
 [3]
 P. B. Kennedy, Fourier series with gaps, Quart. J. Math. Oxford Ser. (2) 7 (1956), 224230. MR 0098272 (20:4733)
 [4]
 , On the coefficients in certain Fourier series, J. London Math. Soc. 33 (1958), 196207. MR 0098274 (20:4735)
 [5]
 , Note on Fourier series with Hadamard gaps, J. London Math. Soc. 39 (1964), 115116. MR 0162087 (28:5288)
 [6]
 M. E. Noble, Coefficient properties of Fourier series with a gap condition, Math. Ann. 128 (1954), 5562. MR 0063469 (16:126a)
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 R. E. A. C. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ. Vol. 19, Amer. Math. Soc., Providence, R.I., 1934. MR 1451142 (98a:01023)
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 J. R. Patadia, On the absolute convergence of lacunary Fourier series, J. London Math. Soc. (2) 14 (1976), 113119. MR 0427936 (55:966)
 [9]
 S. B. Stečkin, On the absolute convergence of orthogonal series. I, Amer. Math. Soc. Transl. (1) 3 (1963), 271280.
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 M. Tomić, On the order of magnitude of Fourier coefficients with Hadamard gaps, J. London Math. Soc. 37 (1962), 117120. MR 0133639 (24:A3465)
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 , On the coefficients of Fourier series with Hadamard gaps, Bull. Acad. Serbe Sci. Arts Cl. Sci. Math. Natur. Sci. Math. 35 (1966), 6368. MR 0208251 (34:8061)
 [12]
 A. Zygmund, Trigonometrical series, Warsaw, 1935.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197804931382
PII:
S 00029939(1978)04931382
Keywords:
Lacunary Fourier series,
absolute convergence,
modulus of continuity,
Hadamard lacunarity condition
Article copyright:
© Copyright 1978
American Mathematical Society
