A note on the absolute convergence of lacunary Fourier series

Authors:
N. V. Patel and V. M. Shah

Journal:
Proc. Amer. Math. Soc. **93** (1985), 433-439

MSC:
Primary 42A55; Secondary 42A28

MathSciNet review:
773997

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Abstract: P. B. Kennedy [**3**] studied lacunary Fourier series whose generating functions are of bounded variation on a subinterval of and satisfy a Lispschitz condition of order on . We show that the conclusion of one of his theorems on the absolute convergence of Fourier series remains valid when the function is merely of bounded th variation in and belongs to a class in . Our results also generalize three theorems of S. M. Mazhar [**4**].

**[1]**G. H. Hardy and J. E. Littlewood,*Some properties of fractional integrals. I*, Math. Z.**27**(1928), no. 1, 565–606. MR**1544927**, 10.1007/BF01171116**[2]**-,*Notes on the theory of series*. IX:*On the absolute convergence of Fourier series*, J. London Math. Soc.**3**(1928), 250-253.**[3]**P. B. Kennedy,*Fourier series with gaps*, Quart. J. Math. Oxford Ser. (2)**7**(1956), 224–230. MR**0098272****[4]**Syed Mohammad Mazhar,*On the absolute convergence of a Fourier series with gaps*, Proc. Nat. Inst. Sci. India Part A**26**(1960), 104–109. MR**0126122****[5]**M. E. Noble,*Coefficient properties of Fourier series with a gap condition*, Math. Ann.**128**(1954), 55–62; correction, 256. MR**0063469****[6]**Raymond E. A. C. Paley and Norbert Wiener,*Fourier transforms in the complex domain*, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR**1451142****[7]**A. Zygmund,*Trigonometrical series*, PWN, Warsaw, 1935.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1985-0773997-X

Keywords:
Lacunary Fourier series,
absolute convergence,
Lipschitz condition,
bounded th variation

Article copyright:
© Copyright 1985
American Mathematical Society