Absolute summability and its application to Fourier series

Abstract

The object of this dissertation is to present results on absolute summability by the methods of Hausdorff, Cesaro, Holder and Abel and to show as far as possible their application in the theory of Fourier Series. Cesaro and Holder absolute summability are treated as special cases of hausdorff absolute summability but the more elementary original definitions are also given. The idea of absolute summability of a series was first introduced in 1911 by Fekete in the case where is a positive integer. The next development was in 1925 when Kogbetlianz proposed a definition for absolute surarnability of order where is any real number other than a negative integer, and developed some of the properties of absolutely summable series giving results analagous to those already found for summable series and some new results on the multiplication of absolutely summable series. Within the last fifteen years the subject has been developed by Bosanquet, Chow, Hyslop, Wang and other writers, particularly with regard to its use in the study of Fourier series and power series.