The Absolute Summability of Series with Applications to Fourier Series

Hyslop, James M (1939) The Absolute Summability of Series with Applications to Fourier Series. DSc thesis, University of Glasgow.

Abstract

This thesis contains a fairly complete account of the modern development of the theory of absolute summability and its applications to Fourier Series. It is necessary to assume a knowledge of the definition and some elementary properties of the Lebesgue and the Stieltjies integrals. I have endeavoured, however, in the introductory chapter to state briefly some results in the theory of integration which are of frequent applicaiton. Chapter 2 contains the definitions of the Cesaro, Riesz and Abel methods of absolute summability, and Chapter 3 sime fundamental theorems, including the consistency theorem for each method and a Tauberian theorem for the Abel method. The equivalence theorem is proved in Chapter 4. The remainer of the thesis is devoted to the absolute summability of Fourier Series. Chapter 5 consists largely of introductory exposition while in Chapter 6 and 7 very general theorems are obtained for a Fourier Series and its allies series respectively. In Chapter 8 the behaviour of the Fourier series of a function satisfying a Lipschitz consition is discussed.

• Item Type: Thesis (DSc)
• Qualification Level: Doctoral
• Keywords: Theoretical mathematics
• Date of Award: 1939
• Depositing User: Enlighten Team
• Unique ID: glathesis:1939-80172
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 02 Mar 2020 23:14
• Last Modified: 02 Mar 2020 23:14
• URI: http://theses.gla.ac.uk/id/eprint/80172