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Stages Up To The Verge Of Summability: A Comprehensive Exploration

Thomas Hardy

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Published in Infinite In A History Of Analysis: Stages Up To The Verge Of Summability (De Gruyter Textbook)

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The concept of summability plays a pivotal role in various branches of mathematics, including analysis, harmonic analysis, and differential equations. Its importance lies in providing a framework for understanding the convergence and divergence of infinite series, which has wide-ranging applications in fields such as numerical analysis, probability theory, and control theory. In this context, the book "Stages Up To The Verge Of Summability" by De Gruyter Textbook offers a comprehensive and in-depth treatment of this fundamental mathematical concept.

Infinite in a History of Analysis: Stages up to the Verge of Summability (De Gruyter Textbook)

Infinite Series in a History of Analysis: Stages up to the Verge of Summability (De Gruyter Textbook)

by David Sinclair

4.1 out of 5

Language	:	English
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Print length	:	143 pages

Chapter 1: The Basics of Summability

This chapter provides an to the fundamental concepts related to summability. It begins with a detailed discussion of convergence and divergence tests for series, including the Cauchy criterion, the ratio test, and the root test. The chapter then introduces various summability methods, such as the Cesàro method, the Abel method, and the Borel method. These methods provide alternative ways to assign values to divergent series, thereby extending the concept of convergence.

Chapter 2: Strong Summability

Chapter 2 delves into the concept of strong summability, which is a more restrictive notion than ordinary summability. Strong summability methods require the convergence of the series to be uniform in some sense. The chapter explores various types of strong summability, including absolute summability, strong Cesàro summability, and strong Borel summability. These methods are particularly useful in studying the convergence of Fourier series and other orthogonal expansions.

Chapter 3: Tauberian Theorems

Chapter 3 introduces Tauberian theorems, which provide connections between the summability of a series and the asymptotic behavior of its terms. These theorems play a crucial role in analyzing the asymptotic behavior of solutions to differential equations and integral equations. The chapter covers classical Tauberian theorems, such as the Hardy-Littlewood theorem and the Wiener-Tauberian theorem, as well as their applications in various areas of mathematics.

Chapter 4: Summability of Fourier Series

Chapter 4 focuses on the summability of Fourier series, which are essential tools in harmonic analysis and other areas of mathematics. It investigates the convergence and divergence of Fourier series using various summability methods. The chapter also explores the relationship between the summability of a Fourier series and the properties of its coefficients. These results have important applications in studying the smoothness and regularity of functions.

Chapter 5: Applications in Differential Equations

Chapter 5 explores the applications of summability in the theory of differential equations. It demonstrates how summability methods can be used to analyze the existence, uniqueness, and asymptotic behavior of solutions to ordinary and partial differential equations. The chapter discusses specific techniques, such as the Laplace transform and the method of steps, which utilize summability to obtain solutions to complex differential equations.

"Stages Up To The Verge Of Summability" by De Gruyter Textbook serves as a comprehensive and authoritative reference for mathematicians, researchers, and students working in areas involving summability. The book's clear and systematic exposition, coupled with its wealth of examples and exercises, makes it an invaluable resource for anyone seeking a deep understanding of this fundamental mathematical concept and its far-reaching applications.

Infinite Series in a History of Analysis: Stages up to the Verge of Summability (De Gruyter Textbook)

by David Sinclair

4.1 out of 5