T · C1 summability of a sequence of Fourier coefficients

摘要

Let Bn(x) denote the nth term of the conjugate series of a Fourier series of function f. Mohanty and Nanda [R. Mohanty, M. Nanda, On the behavior of Fourier coefficients, Proc. Am. Math. Soc. 5 (1954) 79–84] were the first to establish a result for C1-summability of the sequence {nBn(x)}. Varshney [O.P. Varshney, On a sequence of Fourier coefficients, Proc. Am. Math. Soc. 10 (1959) 790–795] improved it for the product summability H1 · C1, which was generalized by various investigators using different summability methods with different set of conditions. In this note, we extend the result of Mittal [M.L. Mittal, On the summability of a sequence of Fourier coefficients, Bull. Cal. Math. Soc. 81 (1989) 25–31], which in turn generalizes the results of Prasad [K. Prasad, On the (N, pn) · C1 summability of a sequence of Fourier coefficients, Indian J. Pure Appl. Math. 12 (7) (1981) 874–881] and Varshney [O.P. Varshney, On a sequence of Fourier coefficients, Proc. Am. Math. Soc. 10 (1959) 790–795].