Absolute Norlund summability of Fourier series

Abstract

In this thesis we examine the properties of the absolute Norlund Summability (ANS) method and its application to Fourier Series of a function f(x), periodic with period 2π and integrable L. In chapter II we discuss the various consistency theorems for ANS. Chapter III consists of a theorem on the ANS of Fourier Series of a function f(x) belonging to the class Lip α. Chapter IV contains a result on the ANS of Fourier Series when ∅(t)= ½{f(x-t)} is of bounded variation in the interval (O, π). In 1959 Varshney [16] proved a theorem on the absolute harmonic summability of a series related to Fourier Series of f(x) when ϕ(t) is of bounded variation in the interval (O,π). In Chapter V we prove a theorem on the ANS of Fourier Series when cp(t) is of bounded variation which generalises the result on absolute harmonic summability due to Varshney and includes the result given in Chapter IV. In Chapter VI we prove a theorem on the ANS of Fourier Series under the condition φ(□(1/t))|ϕ(t)| = 0(1) where φ(n) is a monotonic increasing function.