Absolute Riesz summability of Fourier series. I

Authors:
G. D. Dikshit and C. S. Rees

Journal:
Proc. Amer. Math. Soc. **82** (1981), 231-238

MSC:
Primary 42A28; Secondary 40F05

MathSciNet review:
609657

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove some theorems on the absolute summability of Fourier series which connect diverse results such as Bosanquet's classical theorem (1936), Mohanty (1952), and Ray (1970) and the recent result of Nayak (1971).

It is also shown that in some sense some of the conclusions of the paper are the best possible.

**[1]**L. S. Bosanquet,*Note on the absolute summability**of a Fourier series*, J. London Math. Soc.**11**(1936), 11-15.**[2]**L. S. Bosanquet and H. C. Chow,*Some remarks on convergence and summability factors*, J. London Math. Soc.**32**(1957), 73–82. MR**0083607****[3]**K. Chandrasekharan and S. Minakshisundaram,*Typical means*, Oxford University Press, 1952. MR**0055458****[4]**B. Kuttner,*On the ‘second theorem of consistency’ for absolute Riesz summability*, Proc. London Math. Soc. (3)**29**(1974), 17–32. MR**0352784****[5]**R. Mohanty,*Absolute Cesàro summability of a series associated with a Fourier series*, Bull. Calcutta Math. Soc.**44**(1952), 152–154. MR**0056729****[6]**M. K. Nayak,*On the absolute summability and convergence of Fourier series and associated series*, Proc. Cambridge Philos. Soc.**70**(1971), 421–433. MR**0290031****[7]**B. K. Ray,*On the absolute summability of some series related to a Fourier series*, Proc. Cambridge Philos. Soc.**67**(1970), 29–45. MR**0412719**

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DOI:
https://doi.org/10.1090/S0002-9939-1981-0609657-1

Keywords:
Fourier series,
absolute Riesz summability,
absolute Cesàro summability,
absolute convergence,
function of bounded variation,
totally regular methods

Article copyright:
© Copyright 1981
American Mathematical Society