T . C1 summability of a sequence of Fourier coefficients

Let B-n(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 C-1-summability of the sequence {nB(n)(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 H-1 center dot C-1, 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 parallel to T parallel to . C-1 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,p(n)) . C-1 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]. (C) 2008 Elsevier Inc. All rights reserved.