Product (N,?pn) (C,?1) summability of a sequence of Fourier coefficients

Purpose: The purpose of the present paper is to study the product (N, pn) (C, 1) summability of a sequence of Fourier coefficients which extends a theorem of Prasad. Methods: We use Np. C 1 summability methods with dropping monotonicity on the generating sequence {pn − k} (that is, by weakening the conditions on the filter, we improve the quality of digital filter). Results: Let Bn(x) denote the nth term of conjugate series of a Fourier series. Mohanty and Nanda were the first to establish a result for C1 summability of the sequence {n Bn(x)}. Varshney improved the result for H1. C1 summability which was generalized by various investigators using different summability methods with different sets of conditions. In this paper, we extend a result of Prasad by dropping the monotonicity on the sequence {pn − k}. Conclusions: Various results pertaining to the C1 and H1. C1 summabilities of the sequence {n Bn(x)} have been reviewed and the condition of monotonicity on the means generating the sequence {pn − k} has been relaxed. Moreover, a proper set of conditions have been discussed to rectify the errors pointed out in Remark 3.2 (1) and (2).