Analytic Summability Theory

Type
Book Chapter

Authors
Alabdulmohsin, Ibrahim

KAUST Department
Computer Science Program
King Abdullah University of Science and Technology, Dhahran, Saudi Arabia

Online Publication Date
2018-03-08

Print Publication Date
2018

Date
2018-03-08

Abstract
The theory of summability of divergent series is a major branch of mathematical analysis that has found important applications in engineering and science. It addresses methods of assigning natural values to divergent sums, whose prototypical examples include the Abel summation method, the Cesaro means, and the Borel summability method. As will be established in subsequent chapters, the theory of summability of divergent series is intimately connected to the theory of fractional finite sums. In this chapter, we introduce a generalized definition of series as well as a new summability method for computing the value of series according to such a definition. We show that the proposed summability method is both regular and linear, and that it arises quite naturally in the study of local polynomial approximations of analytic functions. The materials presented in this chapter will be foundational to all subsequent chapters.

Citation
Alabdulmohsin IM (2018) Analytic Summability Theory. Summability Calculus: 65–91. Available: http://dx.doi.org/10.1007/978-3-319-74648-7_4.

Publisher
Springer Nature

Journal
Summability Calculus

DOI
10.1007/978-3-319-74648-7_4

Additional Links
https://link.springer.com/chapter/10.1007%2F978-3-319-74648-7_4

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