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On the Nörlund summability of Fourier series

Published online by Cambridge University Press:  24 October 2008

C. T. Rajagopal
C. T. Rajagopal
Affiliation:
Ramanujan Institute of Mathematics, University of Madras
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Extract

1. The purpose of this note is to prove a result which includes certain classical theorems generally thought of as being unconnected; in explicit terms, a result about the Fourier series of a periodic Lebesgue-integrable function showing that the series is summable at a point by a Nörlund method (N, pn) defined as usual ((2), p. 64) if pn ↓ 0, Σpn = ∞ and the point is in a certain subset of the Lebesgue set. More precisely, the purpose is to prove Theorem I on the Nörlund summability of Fourier series and to derive from it the well-known Theorems A, B which follow and the recent extension of Theorem A in Theorem A' which appears later and is due to Sahney (8).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 1 , January 1963 , pp. 47 - 53
Copyright
Copyright © Cambridge Philosophical Society 1963

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References

REFERENCES

(1)Hardy, G. H. and Littlewood, J. E.Notes on the theory of series (XVIII): On the convergence of Fourier series. Proc. Cambridge Philos. Soc. 31 (1935), 317–323.Google Scholar
(2)Hardy, G. H.Divergent series (Oxford, 1949).Google Scholar
(3)Iyengar, K. S. K.A Tauberian theorem and its application to Fourier series. Proc. Indian Acad. Sci. Sect. A, 18 (1943), 81–87.Google Scholar
(4)Iyengar, K. S. K.New convergence and summability tests for Fourier series. Proc. Indian Acad. Sci. Sect. A, 18 (1943), 113–120.CrossRefGoogle Scholar
(5)Iyengar, K. S. K.Notes on summability. II. On the relation between summability by Nörlund means of a certain type and summability by Valiron means. Half-yearly J. Mysore Univ. Sect. B (N.S.), 4 (1944), 161–166.Google Scholar
(6)Jurkat, W.Zur Konvergenztheorie der Fourier–Reihen. Math. Z. 53 (1950/1951), 309–339.Google Scholar
(7)Prasad, B. N. and Siddiqi, J. A.On the Nörlund summability of the rth derived Fourier series. J. Indian Math. Soc. (N.S.), 14 (1950), 159–170.Google Scholar
(8)Sahney, B. N.On the (H, p) summability of Fourier series. Boll. Un. Math. Ital. (3), 16 (1961), 156–163.Google Scholar
(9)Siddiqi, J. A.On the harmonic summability of Fourier series. Proc. Indian Acad. Sci. Sect. A, 28 (1949), 527–531.Google Scholar
(10)Sinvhal, S. D.Sur la sommabilité (C, 1) de la série de Fourier. Bull. Sci. Math. (2), 79 (1955), 169–173.Google Scholar
(11)Varshney, O. P.On the relation between harmonic summability and summability by Riesz means of a certain type. Toˇhoku Math. J. (2), 11 (1959), 20–24.Google Scholar
(12)Wang, F. T.On Riesz summability of Fourier series. Proc. London Math. Soc. (2), 47 (1942), 308–325.Google Scholar
(13)Zygmund, A.Trigonometrical series (Chelsea reprint; New York, 1952).Google Scholar