Tauberian theorems for certain summability methods

by M. S. Watkins

Publisher: [s.n.] in Toronto

Written in English
Published: Downloads: 126
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Edition Notes

Thesis (M.A.)--University of Toronto, 195-?

StatementM. S. Watkins.
ID Numbers
Open LibraryOL14862282M

COVID Resources. Reliable information about the coronavirus (COVID) is available from the World Health Organization (current situation, international travel).Numerous and frequently-updated resource results are available from this ’s WebJunction has pulled together information and resources to assist library staff as they consider how to handle .   The general framework of a Tauberian theorem is that, given a summability method A and a summability method B, we seek a theorem that states: If a series or integral is summable with method A, and specified extra hypotheses are true, then it is also summable with method B. Part I contains a short general introduction to summability, the basic classical theory concerning mainly inclusion theorems and theorems of the Silverman-Toeplitz type, a presentation of the most important classes of summability methods, Tauberian theorems, and applications of matrix methods.   Pati () established two Tauberian theorems which are more general than a theorem of Pati () and a theorem of Littlewood (). Our aim in this paper is to introduce some new conditions in terms of [[tau].sup.[alpha].sub.n] to recover (C, [alpha]) convergence of ([[tau].sub.n]) from its [(A).sup.(k)](C, [alpha]) summability.

Tauberian Theory: A Century of Developments Grundlehren der mathematischen Wissenschaften: : Korevaar, Jacob: Libros en idiomas extranjerosAuthor: Jacob Korevaar.   Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates. The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early results. summability under certain conditions. As special cases of our main results, we get discrete analogues of some well-known Tauberian theorems in the literature. Key words: Tauberian conditions, discrete summability, power series methods 1. Introduction and preliminaries Let P∞.   This book is primarily about summability, that is, various methods to assign a useful value to a divergent series, usually by forming some kind of mean of the partial summands. methods and the algebraic view of Wiener’s general Tauberian theorem. The treatment is specific rather than general, and each summability method is handled.

Description: This book presents results about certain summability methods, such as the Abel method, the Norlund method, the Weighted mean method, the Euler method and the Natarajan method, which have not appeared in many standard books. It proves a few results on the Cauchy multiplication of certain summable series and some product theorems. Tauberian Theorems - I Tauberian Theorems - II Relationships with other methods Applications of Borel's Methods-- References. (source: Nielsen Book Data) Summability methods are transformations that map sequences (or functions) to sequences (or functions). A prime requirement for a "good" summability method is that it preserves. Alternative forms []. tauberian theorem; Etymology []. After Austrian and Slovak mathematician Alfred Tauber ().. Noun []. Tauberian theorem (plural Tauberian theorems) (mathematical analysis) Any of a class of theorems which, for a given Abelian theorem, specifies conditions such that any series whose Abel sums converge (as stipulated by the Abelian theorem.

Tauberian theorems for certain summability methods by M. S. Watkins Download PDF EPUB FB2

Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates. The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early : Hardcover.

An introductory course in summability theory for students, researchers, physicists, and engineers. Tauberian theorems for certain summability methods book In creating this book, the authors’ intent was to provide graduate students, researchers, physicists, and engineers with a reasonable introduction to summability by: 4.

Following a brief introduction to the history of summability theory, general matrix methods are introduced, and the Silverman-Toeplitz theorem on regular matrices is discussed. A variety of special summability methods, including the Nörlund method, the Weighted Mean method, the Abel method, and the (C, 1) - method are next examined.

About this book. Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates.

The author shows Tauberian theorems for certain summability methods book development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early : Springer-Verlag Berlin Heidelberg. The aim of this paper is to introduce some new Tauberian conditions for (A)(C,α) summability method using the concept of the general control modulo of the oscillatory behavior of the integer order defined by Dik [7] and to extend and generalize some of the results of Pati [8].Cited by: 9.

Silverman-type theorems, consistency theorems and the like. In Chapters 2 and 3 we discuss matrix methods like Ces`aro methods, Hausdorff methods and others, and power series methods like the Abel method and the Borel method.

Chapter 4 deals with Tauberian theorems for certain (classes of) summability methods. Some applications are given in. Kratz and Stadtmüller obtained Tauberian conditions for (J, p) summability method under certain conditions on (p n). Tauberian theorem for (N ¯, p) summability method is examined by several authors such as Hardy, Ananda-Rau, Tietz, Tietz and Zeller, and Móricz and by: A variety of special summability methods, including the Nörlund method, the Weighted Mean method, the Abel method, and the (C, 1) – method are next examined.

An entire chapter is devoted to a discussion of some elementary Tauberian theorems involving certain summability methods. theorems of Tauberian type Theorems establishing conditions which determine the set of series (or sequences) on which for two given summation methods and the inclusion holds.

Most frequently in the theory of summation, the case in which method. Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates.

The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early results. Summability Theory: Convergence Tests, Divergent Series, Sequence Spaces, Summability Methods, Tauberian Theorems, Harmonic Series.

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. About this book. Introduction. Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates.

The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early results.

such converse theorems, and they coined the term Tauberian to describe them. In summability language Theorem T can be expressed as: If X∞ n=0 an = `(A),where Adenotes the Abel summability method, and if the Tauberian condition (T0) holds, then X∞ n=0 an = `. The simplest example of an Abel summable series that is not convergent is given by.

The latter papers contain certain o - Tauberian theorems for all power series methods in question and O-Tauberian theorems, if the weight sequence (p n) can be interpolated by alogarithmico-exponential function g ()(see e.g.

[4]), by: Exercise Show Tauber’s theorem. Other Summability Methods We can de ne other summability methods in the same spirit as that of Abel summability.

Let ’be a function such that ’(0) = 1, lim x!1’(x) = 0, and its derivative ’0is integrable on the interval [0;1).

We say that the series P 1 n=0 c nis (’) summable to if there is 0 File Size: KB. Some Tauberian theorems for the weighted mean methods of summability Article in Computers & Mathematics with Applications 62(6) September with 60 Reads How we measure 'reads'.

A summability method M is regular if it agrees with the actual limit on all convergent series. Such a result is called an Abelian theorem for M, from the prototypical Abel's theorem.

More interesting, and in general more subtle, are partial converse results, called Tauberian theorems, from a prototype proved by Alfred Tauber.

Some suitable conditions given in terms of the oscillatory behavior of (V (0) n (u)) are used to be Tauberian conditions for certain summability methods in. A variety of special summability methods, including the Norlund method, the Weighted Mean method, the Abel method, and the (C, 1) - method are next examined.

An entire chapter is devoted to a discussion of some elementary Tauberian theorems involving certain summability : Ants Aasma, Hemen Dutta, P. Natarajan. 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. We investigate conditions under which a sequence that belongs to a summability field of certain power series methods of Abel- or Borel-type is contained in the summability field of some Cesáro method as well.

The Tauberian condition presented here is weaker than those given for special Borel-type methods in the : W Motzer, U Stadtmüller. Description: This book discusses recent developments in and contemporary research on summability theory, including general summability methods, direct theorems on summability, absolute and strong summability, special methods of summability, functional analytic methods in summability, and related topics and applications.

All contributing authors are eminent. 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.

Abel, Abel summability, summability methods for integrals See also: Annotations for § summability methods for integrals Notes: See Miller and Ross and Andrews et al.

(, pp.–). Referenced by: Tauberian Theorems. Canak and Totur [8] proved a Tauberian theorem for Cesaro summability methods, and Totur and Dik [20] gave some one-sided Tauberian conditions for a general summability method using the general control modulo of integer order.

Moreover, various Tauberian theorems have been demonstrated by Canak [1,2,3,4] and Canak et al. [10]. of functional analysis. From this, inIkehara deduced a Tauberian theorem for Dirichlet series (now known as the Wiener-Ikehara Theorem), with which one can give simple proofs of the Prime Number Theorem and various generalizations thereof.

InNewman published a new method to derive Tauberian theorems,File Size: KB. A variety of special summability methods, including the Nörlund method, the Weighted Mean method, the Abel method, and the (C, 1) - method are next examined.

An entire chapter is devoted to a discussion of some elementary Tauberian theorems involving certain summability methods.

A Tauberian theorem for the generalized Nörlund summability method;Necessary and sufficient Tauberian conditions for certain weighted mean methods of summability, Acta Math.

Hungar. 66 (), no. 1–2, – Crossref Google Scholar Georgian Mathematical Journal, ISSN (Online)Author: İbrahim Çanak, Naim L. Braha, Ümit Totur. Summability is a mathematical topic with a long tradition and many applications in, for example, function theory, number theory, and stochastics.

It was originally based on classical analytical methods, but was strongly influenced by modern functional analytical methods during the last seven decades.

The present book aims to introduce the reader to the wide field of summability Reviews: 1. In [] Robison presented the following notion of regularity for four-dimensional matrix transformation and a Silverman-Toeplitz type characterization of such a tion The four-dimensional matrix A is said to be RH-regular if it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit.

The assumption of Author: Fatih Nuray, Fatih Nuray, Richard F Patterson. Tauberian Theorem If an infinite series is summable or an integral is integrable by a certain method, it is of interest to know under what conditions summability or integrability may be obtained for a weaker method (see).

The conditions that must be imposed on the series or integral are established by theorems known as Tauberian theorems. One of the.Get this from a library! Tauberian theory: a century of developments. [Jacob Korevaar] -- "Tauberian theory compares different summability methods both for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates.

The author shows.springer, 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.