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On the Behaviour of a Series Associated with the Conjugate Series of a Fourier Series

  • R. Mohanty (a1) and B. K. Ray (a1)

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1.1. Definition. Let λ ≡ λ(ω) be continuous, differentiable, and monotonic increasing in (0, ∞) and let it tend to infinity as ω → ∞. Suppose that ∑ an (we write ∑ for throughout the present paper) is a given infinite series and let

The series ∑ an is said to be summable |R, λ, r|, where r > 0, if

where A is a fixed positive number (6, Definition B). Now, for r > 0, m < ω < m + 1,

Hence, ∑ an is summable |R, λ, r|, where r > 0, if

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References

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1. Bosanquet, L. S. and Hyslop, J. M., On the absolute summability of the allied series of a Fourier series, Math. Z. 42 (1937), 489512.
2. Chandrasekharan, K., The second theorem of consistency for absolutely summable series, J. Indian Math. Soc. (N.S.) 6 (1942), 168180.
3. Hardy, G. H., Divergent series, p. 87 (Oxford, at the Clarendon Press, 1963).
4. Matsumoto, K., On the absolute Cesàro summability of a series related to a Fourier series, Tôhoku Math. J. 8 (1956), 205222.
5. Misra, M. L., On the determination of the jump of a function by its Fourier co-efficients, Quart. J. Math. Oxford Ser. 18 (1947), 147156.
6. Mohanty, R., On the absolute Riesz summability of Fourier series and an allied series, Proc. London Math. Soc. (2) 52 (1951), 295320.
7. Mohanty, R., On the summability \R, log co, 1| of a Fourier series, J. London Math. Soc. 25 (1950), 6772.
8. Mohanty, R., On the convergence factor of a Fourier series, Proc. Cambridge Philos. Soc. 63 (1967), 129131.
9. Mohanty, R. and Mahapatra, S., On the absolute logarithmic summability of a Fourier series and its differentiated series, Proc. Amer. Math. Soc. 7 (1956), 254259.
10. Obrechkoff, N., Sur la sommation des séries de Dirichlet, C. R. Acad. Sci. Paris 186 (1928), 215217.
11. Obrechkoff, N., Sur la sommation des séries trigonométriques de Fourier par les moyennes arithmétiques, Bull. Soc. Math. France 62 (1934), 84109; 167-184.
12. Zygmund, A., Trigonometric series, 2nd éd., Vol. I, p. 52 (Cambridge Univ. Press, New York, 1959).
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On the Behaviour of a Series Associated with the Conjugate Series of a Fourier Series

  • R. Mohanty (a1) and B. K. Ray (a1)

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