On asymptotic behaviours of trigonometric series with δ-Quasi-monotone coefficients

Abstract

LetThe asymptotic behaviours of ƒ(x) and g(x), as x→+0, were first given by G. H. Hardy in (4), (5). In his papers {an}; is a monotone decreasing sequence. Further results on the asymptotic behaviours of ƒ(x) and g(x), as x→+0, for monotone coefficients have been given in (9) and (1). Recently, the results have been generalized to quasi-monotone coefficients.This paper is concerned with asymptotic behaviours of ƒ(x) and g(x) for δ-quasi-monotone coefficients.In what follows, we shall denote by L(x) a slowly varying function in the sense of Karamata (6), i.e.