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. It also proves a number of Steinhaus type theorems. In addition, it introduces a new definition of convergence of a double sequence and double series and proves the Silverman-Toeplitz theorem for four-dimensional infinite matrices, as well as Schur's and Steinhaus theorems for four-dimensional infinite matrices. The Norlund method, the Weighted mean method and the Natarajan method for double sequences are also discussed in the context of the new definition. Divided into six chapters, the book supplements the material already discussed in G.H.Hardy's Divergent Series. It appeals to young researchers and experienced mathematicians who wish to explore new areas in Summability Theory..
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1 General Summability Theory and Steinhaus Type Theorems |
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1 | (26) |
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1.1 Basic Definitions and Concepts |
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1 | (2) |
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1.2 The Silverman-Toeplitz Theorem, Schur's Theorem, and Steinhaus Theorem |
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3 | (10) |
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1.3 A Steinhaus Type Theorem |
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13 | (3) |
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1.4 The Role Played by the Sequence Spaces Ar |
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16 | (3) |
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1.5 More Steinhaus Type Theorems |
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19 | (8) |
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25 | (2) |
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2 Core of a Sequence and the Matrix Class (l, l) |
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27 | (10) |
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27 | (1) |
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2.2 Natarajan's Theorem and Knopp's Core Theorem |
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28 | (3) |
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2.3 Some Results for the Matrix Class (l, l) |
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31 | (4) |
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35 | (2) |
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36 | (1) |
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3 Special Summability Methods |
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37 | (26) |
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37 | (11) |
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3.2 (M, λn) Method or Natarajan Method |
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48 | (6) |
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3.3 The Abel Method and the (M, λn) Method |
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54 | (1) |
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3.4 The Euler Method and the (M, λn) Method |
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55 | (8) |
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61 | (2) |
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4 More Properties of the (M, λn) Method and Cauchy Multiplication of Certain Summable Series |
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63 | (20) |
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4.1 Some Nice Properties of the (M, λn) Method |
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63 | (6) |
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4.2 Iteration of (M, λn) Methods |
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69 | (4) |
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4.3 Cauchy Multiplication of (M, λn)-Summable Series |
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73 | (4) |
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4.4 Cauchy Multiplication of Euler Summable Series |
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77 | (6) |
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82 | (1) |
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5 The Silverman-Toeplitz, Schur, and Steinhaus Theorems for Four-Dimensional Infinite Matrices |
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83 | (18) |
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5.1 A New Definition of Convergence of a Double Sequence and a Double Series |
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83 | (2) |
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5.2 The Silverman-Toeplitz Theorem for Four-Dimensional Infinite Matrices |
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85 | (9) |
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5.3 The Schur and Steinhaus Theorems for Four-Dimensional Infinite Matrices |
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94 | (7) |
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100 | (1) |
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6 The Norlund, Weighted Mean, and (M, λm,n) Methods for Double Sequences |
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101 | (28) |
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6.1 Norlund Method for Double Sequences |
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101 | (6) |
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6.2 Weighted Mean Method for Double Sequences |
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107 | (13) |
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6.3 (M, λm,n) or Natarajan Method for Double Sequences |
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120 | (9) |
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128 | (1) |
| Index |
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129 | |
P.N. NATARAJAN, formerly with the Department of Mathematics, Ramakrishna Mission Vivekananda College, Chennai, India, has been an independent researcher and mathematician since his retirement in 2004. He did his Ph.D. at the University of Madras, under Prof. M.S. Rangachari, former director and head of the Ramanujan Institute for Advanced Study in Mathematics, University of Madras. An active researcher, Prof. Natarajan has published over 100 research papers in several international journals like the Proceedings of the American Mathematical Society, Bulletin of the London Mathematical Society, Indagationes Mathematicae, Annales Mathematiques Blaise Pascal, and Commentationes Mathematicae (Prace Matematyczne). His research interests include summability theory and functional analysis (both classical and ultrametric). Professor Natarajan was honored with the Dr. Radhakrishnan Award for the Best Teacher in Mathematics for the year 199091 by the Government of Tamil Nadu. In addition to being invited to visit several renowned institutes in Canada, France, Holland and Greece Prof. Natarajan has participated in several international conferences and chaired sessions. He has authored two books, An Introduction to Ultrametric Summability Theory and its second edition, both published with Springer in 2013 and 2015, respectively.
Lisainfo e-raamatute kohta