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On the convergence factor of a Fourier series and a differentiated Fourier series

Published online by Cambridge University Press:  24 October 2008

R. Mohanty
Affiliation:
Ravenshaw College, Cuttack-3, India
B. K. Ray
Affiliation:
Ravenshaw College, Cuttack-3, India

Extract

Definition A. The serieswith partial sumUn (or the sequence {Un}) is said to be summable by logarithmic means to the sum U, if

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1969

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References

REFERENCES

(1)Hardy, G. H.On the summability of Fourier series. Proc. London Math Soc. 12 (1913), p. 365–72.CrossRefGoogle Scholar
(2)Hardy, G. H.Divergent series, p. 87 (Oxford, 1963).Google Scholar
(3)Hardy, G. H.The summability of a Fourier series by logarithmic means. Quart. J. Math. Oxford ser. 2 (1931), 107–12.CrossRefGoogle Scholar
(4)Hardy, G. H. & Riesz, M.The general theory of Dirichlet's series (Cambridge, 1915).Google Scholar
(5)Misra, M. L.On the determination of the jump of a function by its Fourier coefficients. Quart. J. Math. Oxford ser. 18 (1947), 147156.CrossRefGoogle Scholar
(6)Mohanty, R.On the convergence factor of a Fourier series. Proc. Cambridge Philos. Soc. (1967), 63, 129131.CrossRefGoogle Scholar
(7)Mohanty, R. & Nanda, M.The summability by logarithmic means of the derived Fourier series. Quart. J. Math. Oxford ser. 6 (1955), 53.CrossRefGoogle Scholar
(8)Mohanty, R. & Nanda, M.On the logarithmic means of the derived conjugate series of a Fourier series. Proc Amer. Math. Soc. 7 (1956), 397400.CrossRefGoogle Scholar
(9)Sunouchi, G.Notes on Fourier Analysis XXXIX. Tôhoku Math. J. 3 (1951), 7188, Theorem 2.1.CrossRefGoogle Scholar
(10)Zygmund, A.Trigonometric series, vol. I p. 66 (Cambridge University Press, 1959).Google Scholar