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On the strong Nörlund summability of conjugate Fourier series. (English) Zbl 1228.42006

Summary: In continuation of a recent work by M. L. Mittal [J. Math. Anal. Appl. 314, No. 1, 75–84 (2006; Zbl 1080.42001)], the present authors obtain a sufficient condition for the summability \([N,p^{(1)}_n,2]\) of the conjugate Fourier series. In conjunction with the known Tauberian theorem on strong Nörlund summability, which was also considered earlier by M. L. Mittal [J. Indian Math. Soc., New Ser. 44, 369–377 (1980; Zbl 0616.40004)], our result gives a sufficient condition for the summability \([C,1,2]\) of the conjugate Fourier series. Our main theorem generalizes the results given earlier by G. Prasad [On Nörlund summability of Fourier series, Ph.D. thesis, University of Roorkee, Roorkee (1967)] and U. N. Singh [“On the strong summability of a Fourier series and its conjugate series”, Proc. Natl. Inst. Sci. India 13, 319–325 (1947; Zbl 1228.42007)].

MSC:

42A24 Summability and absolute summability of Fourier and trigonometric series
40F05 Absolute and strong summability
Full Text: DOI

References:

[1] Hyslop, J. M., Note on the strong summability of series, Proc. Glasgow Math. Assoc., 1, 16-20 (1952) · Zbl 0049.32103
[2] Mittal, M. L., A Tauberian theorem on strong Nörlund summability, J. Indian Math. Soc. (N.S.), 44, 369-377 (1980) · Zbl 0616.40004
[3] Mittal, M. L., On strong Nörlund summability of Fourier series, J. Math. Anal. Appl., 314, 75-84 (2006) · Zbl 1080.42001
[4] Mittal, M. L.; Kumar, R., A note on strong Nörlund summability, J. Math. Anal. Appl., 199, 312-322 (1996) · Zbl 0858.40010
[5] G. Prasad, On Nörlund summability of Fourier Series, Ph.D. thesis, University of Roorkee, Roorkee, 1967.; G. Prasad, On Nörlund summability of Fourier Series, Ph.D. thesis, University of Roorkee, Roorkee, 1967.
[6] Singh, U. N., On the strong summability of a Fourier series and its conjugate series, Proc. Nat. Inst. Sci. India Part A, 13, 319-325 (1947) · Zbl 1228.42007
[7] Zygmund, A., Trigonometric Series (1959), Cambridge University Press: Cambridge University Press Cambridge · JFM 58.0296.09
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