An introduction to ultrametric summability theory.

*(English)*Zbl 1284.40001
SpringerBriefs in Mathematics. New Delhi: Springer (ISBN 978-81-322-1646-9/pbk; 978-81-322-1647-6/ebook). ix, 102 p. (2014).

An ultrametric space \(X\) is a special kind of metric space in which the inequality
\[
d(x,z)\leq \max\{d(x,y),d(y,z)\}
\]
holds instead of the usual triangle inequality \(d(x,z)\leq d(x,y)+d(y,z)\), for all \(x,y,z \in X.\) Let \(K\) be a field. Specially, consider a mapping \(|.|:K\to \mathbb R\) satisfying the inequality
\[
|x+y|\leq \max\{|x|,|y|\} \tag{*}
\]
instead of the usual triangle inequality for \(x,y\in K.\) If members of \(K\) satisfy the inequality \((*)\), then the field \(K\) is called a non-Archimedean field or ultrametric field. The study of analysis in non-Archimedean fields is known as \(p\)-adic analysis or ultrametric analysis.

This book deals with ultrametric analysis and its applications to summability theory, and consists of four chapters.

Chapter 1 introduces the reader to some topological and geometric properties in ultrametric fields.

In Chapter 2, the author investigates the concepts of differentiability and derivatives in ultrametric analysis and briefly expresses differences between usual calculus and ultrametric calculus.

In Chapter 3, ultrametric Banach spaces are introduced. The author shows that the Hahn-Banach theorem fails to hold in these spaces. Moreover, the classical convexity does not work and it is replaced by the notion of \(K\)-convexity.

Chapter 4 is a main chapter of this book. In this chapter, the author presents a brief survey of the literature on ultrametric summability theory. The celebrated “Silverman-Toeplitz theorem” is proved. Some well-known summability methods like the Nörlund method, the weighted mean method, the \(Y\)-method, the \(M\)-method, the Euler method, and the Taylor method are introduced and their proporties are extensively discussed, and some product theorems and Tauberian theorems are presented. Moreover, double sequences and double series are investigated. A Silverman-Toeplitz theorem for four-dimensional matrices are given and its applications are studied.

This book deals with ultrametric analysis and its applications to summability theory, and consists of four chapters.

Chapter 1 introduces the reader to some topological and geometric properties in ultrametric fields.

In Chapter 2, the author investigates the concepts of differentiability and derivatives in ultrametric analysis and briefly expresses differences between usual calculus and ultrametric calculus.

In Chapter 3, ultrametric Banach spaces are introduced. The author shows that the Hahn-Banach theorem fails to hold in these spaces. Moreover, the classical convexity does not work and it is replaced by the notion of \(K\)-convexity.

Chapter 4 is a main chapter of this book. In this chapter, the author presents a brief survey of the literature on ultrametric summability theory. The celebrated “Silverman-Toeplitz theorem” is proved. Some well-known summability methods like the Nörlund method, the weighted mean method, the \(Y\)-method, the \(M\)-method, the Euler method, and the Taylor method are introduced and their proporties are extensively discussed, and some product theorems and Tauberian theorems are presented. Moreover, double sequences and double series are investigated. A Silverman-Toeplitz theorem for four-dimensional matrices are given and its applications are studied.

Reviewer: Umit Totur (Aydin)

##### MSC:

40-02 | Research exposition (monographs, survey articles) pertaining to sequences, series, summability |

40Cxx | General summability methods |

40Gxx | Special methods of summability |

40A30 | Convergence and divergence of series and sequences of functions |

26E30 | Non-Archimedean analysis |

12J25 | Non-Archimedean valued fields |

46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |