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On the Absolute Cesaro Summability of Negative Order 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. Definition. Let λ ≡ λ(ω) be continuous, differentiable, and monotonie increasing in (0, ∞) and let it tend to infinity as ω → ∞. A series an is summable |R, λ, r|, where r > 0, if

where A is a fixed positive number (6, Definition B).

Let f(t) be a periodic function with period 2π and Lebesgue integrable over (–π, π) and let

1.1

The series conjugate to (1.1), at t = x, is

1.2

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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., Notes on some points in the integral calculus. LXVII: The arithmetic mean of a Fourier constant, Messenger of Math. 58 (1928), 5052.
4. Hyslop, J. M., The absolute summability of series by Rieszian means, Proc. Edinburgh Math. Soc. (2) 5 (1936), 4654.
5. Mazhar, S. M., A Tauberian theorem for absolute summability, Indian J. Math. 1 (1959), 6976.
6. Mohanty, R., On the absolute Riesz summability of Fourier series and allied series, Proc. London Math. Soc. (2) 52 (1951), 295320.
7. Mohanty, R. and Misra, B., On absolute logarithmic summability of a sequence related to a Fourier series, Tôhoku Math. J. (2) 6 (1954), 512.
8. 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.
9. Mohanty, R. and Ray, B. K., On the behaviour of a series associated with the conjugate series of a Fourier series, Can. J. Math. 21 (1969), 535551.
10. Varshney, O. P., On the absolute harmonic summability of a series related to a Fourier series, Proc. Amer. Math. Soc. 10 (1959), 784789.
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On the Absolute Cesaro Summability of Negative Order of a Series Associated with the Conjugate Series of a Fourier Series

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

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