On the strong Nörlund summability of conjugate Fourier series

摘要

In continuation of a recent work by Mittal [M.L. Mittal, On strong Nörlund summability of Fourier series, J. Math. Anal. Appl. 314 (2006),75–84], the present authors obtain a sufficient condition for the summability [N,pn(1),2] of the conjugate Fourier series. In conjunction with the known Tauberian theorem on the strong Nörlund summability, which was also considered earlier by Mittal [M.L. Mittal, A Tauberian theorem on strong Nörlund summability, J. Indian Math. Soc. (N.S.) 44 (1980), 369–377], 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 Prasad [G. Prasad, On Nörlund Summability of Fourier Series, Ph.D. thesis, University of Roorkee, Roorkee, 1967] and Singh [U.N. Singh, On the strong summability of a Fourier series and its conjugate series, Proc. Nat. Inst. Sci. India Part A 13 (1947), 319–325].