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Über dieses Buch

This short monograph is the first book to focus exclusively on the study of summability methods, which have become active areas of research in recent years. The book provides basic definitions of sequence spaces, matrix transformations, regular matrices and some special matrices, making the material accessible to mathematicians who are new to the subject. Among the core items covered are the proof of the Prime Number Theorem using Lambert's summability and Wiener's Tauberian theorem, some results on summability tests for singular points of an analytic function, and analytic continuation through Lototski summability. Almost summability is introduced to prove Korovkin-type approximation theorems and the last chapters feature statistical summability, statistical approximation, and some applications of summability methods in fixed point theorems.

Inhaltsverzeichnis

Chapter 1. Toeplitz Matrices

Abstract
The theory of matrix transformations deals with establishing necessary and sufficient conditions on the entries of a matrix to map a sequence space X into a sequence space Y. This is a natural generalization of the problem to characterize all summability methods given by infinite matrices that preserve convergence.
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Chapter 2. Lambert Summability and the Prime Number Theorem

Abstract
The prime number theorem (PNT) was stated as conjecture by German mathematician Carl Friedrich Gauss (1777–1855) in the year 1792 and proved independently for the first time by Jacques Hadamard and Charles Jean de la Vallée-Poussin in the same year 1896. The first elementary proof of this theorem (without using integral calculus) was given by Atle Selberg of Syracuse University in October 1948. Another elementary proof of this theorem was given by Erdös in 1949.
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Chapter 3. Summability Tests for Singular Points

Abstract
A point at which the function f(z) ceases to be analytic, but in every neighborhood of which there are points of analyticity is called singular point of f(z).
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Chapter 4. Lototski Summability and Analytic Continuation

Abstract
Analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example, in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
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Chapter 5. Summability Methods for Random Variables

Abstract
Let (X k ) be a sequence of independent, identically distributed (i.i.d.) random variables with E | X k | < and EX k = μ, k = 1, 2, . Let A = (a nk ) be a Toeplitz matrix, i.e., the conditions (1.3.1)–(1.3.3) of Theorem 1.3.3 are satisfied by the matrix A = (a nk ). Since
$$\displaystyle\begin{array}{rcl} E\sum _{k=1}^{\infty }\vert a_{ nk}X_{k}\vert = E\vert X_{k}\vert \sum _{k=1}^{\infty }\vert a_{ nk}\vert \leq ME\vert X_{k}\vert,& & {}\\ \end{array}$$
the series $$\sum _{k=0}^{\infty }a_{nk}X_{k}$$ converges absolutely with probability one.
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Chapter 6. Almost Summability

Abstract
In the theory of sequence spaces, an application of the well-known Hahn-Banach Extension Theorem gives rise to the notion of Banach limit which further leads to an important concept of almost convergence. That is, the lim functional defined on c can be extended to the whole of and this extended functional is known as the Banach limit . In 1948, Lorentz  used this notion of weak limit to define a new type of convergence, known as the almost convergence. Since then a huge amount of literature has appeared concerning various generalizations, extensions, and applications of this method.
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Chapter 7. Almost Summability of Taylor Series

Abstract
In Chap. 4, we applied the generalized Lototski or [F, d n ]-summability to study the regions in which this method sums a Taylor series to the analytic continuation of the function which it represents. In the applications of summability theory to function theory it is important to know the region in which the sequence of partial sums of the geometric series is A-summable to $$1/(1 - z)$$ for a given matrix A. The well-known theorem of Okada  gives the domain in which a matrix A = (a jk ) sums the Taylor series of an analytic function f to one of its analytic continuations, provided that the domain of summability of the geometric series to $$1/(1 - z)$$ and the distribution of the singular points of f are known. In this chapter, we replace the [F, d n ]-matrix or the general Toeplitz matrix by almost summability matrix to determine the set on which the Taylor series is almost summable to f(z) (see ). Most of the basic definitions and notations of this chapter are already given in Chap. 4; in fact, this chapter is in continuation of Chap. 4.
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Chapter 8. Matrix Summability of Fourier and Walsh-Fourier Series

Abstract
In this chapter we apply regular and almost regular matrices to find the sum of derived Fourier series, conjugate Fourier series, and Walsh-Fourier series (see  and ). Recently, Móricz  has studied statistical convergence of sequences and series of complex numbers with applications in Fourier analysis and summability.
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Chapter 9. Almost Convergence in Approximation Process

Abstract
Several mathematicians have worked on extending or generalizing the Korovkin’s theorems in many ways and to several settings, including function spaces, abstract Banach lattices, Banach algebras, Banach spaces, and so on. This theory is very useful in real analysis, functional analysis, harmonic analysis, measure theory, probability theory, summability theory, and partial differential equations. But the foremost applications are concerned with constructive approximation theory which uses it as a valuable tool. Even today, the development of Korovkin-type approximation theory is far from complete. Note that the first and the second theorems of Korovkin are actually equivalent to the algebraic and the trigonometric version, respectively, of the classical Weierstrass approximation theorem . In this chapter we prove Korovkin type approximation theorems by applying the notion of almost convergence and show that these results are stronger than original ones.
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Chapter 10. Statistical Summability

Abstract
There is another notion of convergence known as the statistical convergence which was introduced by Fast  and Steinhaus  independently in 1951. In , Moricz mentioned that Henry Fast first time had heard about this concept from Steinhaus, but in fact it was Antoni Zygmund who proved theorems on the statistical convergence of Fourier series in the first edition of his book [101, pp. 181–188] where he used the term “almost convergence” in place of statistical convergence and at that time this idea was not recognized much. Since the term “almost convergence” was already in use (as described earlier in this book), Fast had to choose a different name for his concept and “statistical convergence” was most suitable. In this chapter we study statistical convergence and some of its variants and generalizations. Active researches were started after the paper of Fridy  and since then many of its generalizations and variants have appeared so far, e.g., [38, 62, 64, 70, 74, 76, 77], and so on.
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Chapter 11. Statistical Approximation

Abstract
In the last chapter we discussed statistical summability and its various generalizations and variants, e.g., lacunary statistical convergence, λ-statistical convergence, A-statistical convergence, statistical summability (C, 1), and statistical A-summability. In this chapter, we demonstrate some applications of these summability methods in proving Korovkin-type approximation theorems. Such a method was first used by Gadjiev and Orhan  in which the statistical version of Korovkin approximation was proved by using the test functions 1, x, and x 2. Since then a large amount of work has been done by applying statistical convergence and its variants, e.g., [61, 71–73, 75, 92] for different set of test functions. In this chapter we present few of them and demonstrate the importance of using these new methods of summability.
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Chapter 12. Applications to Fixed Point Theorems

Abstract
Let E be a closed, bounded, convex subset of a Banach space X and f: E ⟶ E. Consider the iteration scheme defined by $$\bar{x_{0}} = x_{0} \in E$$, $$\bar{x}_{n+1} = f(x_{n}),\ x_{n} =\sum \limits _{ k=0}^{n}a_{nk}\bar{x_{k}},\ n \geq 1$$, where A is a regular weighted mean matrix. For particular spaces X and functions f we show that this iterative scheme converges to a fixed point of f. During the past few years several mathematicians have obtained fixed point results using Mann and other iteration schemes for certain classes of infinite matrices. In this chapter, we present some results using such schemes which are represented as regular weighted mean methods. Results of this chapter appeared in [20, 40, 82] and .
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