Classical and modern methods in summability

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Preface......Page 8
Acknowledgements......Page 11
Contents......Page 12
PART I CLASSICAL METHODS IN SUMMABILITY AND APPLICATIONS......Page 16
1.1 The early history of summability-the devil\'s invention......Page 18
1.2 Summability methods: definition and examples......Page 20
1.3 Questions and basic notions......Page 35
1.4 Notes on Chapter 1......Page 40
2 Matrix methods: basic classical theory......Page 41
2.1 Dealing with infinite series......Page 42
2.2 Dealing with infinite matrices......Page 49
2.3 Conservative matrix methods......Page 54
2.4 Coercive and strongly conservative matrix methods......Page 66
2.5 Abundance within domains; factor sequences......Page 76
2.6 Comparison and consistency theorems......Page 90
2.7 Triangles of type M......Page 97
2.8 The mean value property......Page 102
2.9 Potent matrix methods......Page 107
2.10 Notes on Chapter 2......Page 112
3 Special summability methods......Page 114
3.1 Cesaro and Holder methods......Page 115
3.2 Weighted means, Riesz methods......Page 127
3.3 Norlund methods......Page 141
3.4 Hausdorff methods......Page 151
3.5 Methods of function theoretical type......Page 167
3.6 Summability methods defined by power series......Page 172
3.7 Notes on Chapter 3......Page 180
4 Tauberian theorems......Page 182
4.1 Tauberian theorems for Cesaro methods......Page 183
4.2 Tauberian theorems for Riesz methods......Page 193
4.3 Tauberian theorems for power series methods......Page 201
4.4 Hardy--Littlewood\'s D-theorems for the Abel method......Page 206
4.5 Hardy-Littlewood\'s ID-theorem for the Borel method......Page 211
4.6 Notes on Chapter 4......Page 219
5 Application of matrix methods......Page 220
5.1 Boundary behaviour of power series......Page 221
5.2 Analytic continuation......Page 229
5.3 Numerical solution of systems of linear equations......Page 243
5.4 Fourier effectiveness of matrix methods......Page 259
5.5 Notes on Chapter 5......Page 271
PART II FUNCTIONAL ANALYTIC METHODS IN SUMMABILITY......Page 274
6 Functional analytic basis......Page 276
6.1 Topological spaces......Page 277
6.2 Semi-metric spaces......Page 283
6.3 Semi-normed spaces, Banach spaces......Page 295
6.4 Locally convex spaces......Page 307
6.5 Continuous linear maps and the dual space of a locally convex space......Page 321
6.6 Dual pairs and compatible topologies......Page 329
6.7 Frechet spaces......Page 342
6.8 Barrelled spaces......Page 350
7 Topological sequence spaces: K- and FK-spaces......Page 353
7.1 Sequence spaces and their c-duals......Page 354
7.2 K-spaces......Page 364
7.3 FK-spaces......Page 374
7.4 Functional analytic proofs of some Toeplitz-Silverman-type theorems......Page 383
7.5 The dual of FK-spaces......Page 390
7.6 Distinguished subspaces of FK-spaces......Page 400
7.7 Notes on Chapter 7......Page 408
8 Matrix methods: structure of the domains......Page 411
8.1 Domains of matrix methods as FK-spaces......Page 412
8.2 Distinguished subspaces of domains......Page 421
8.3 Replaceability and it-uniqueness of matrices......Page 434
8.4 Examples......Page 438
8.5 Bounded divergent sequences in the domain......Page 453
8.6 Consistency and perfectness......Page 458
8.7 Replaceability and invariance......Page 463
8.8 Notes on Chapter 8......Page 469
PART III COMBINING CLASSICAL AND FUNCTIONAL ANALYTIC METHODS......Page 472
9 Consistency of matrix methods......Page 474
9.1 Consistency and theorems of Mazur-Orlicz type......Page 475
9.2 p-bounded sequences and domains......Page 490
9.3 p-consistency and p-comparison......Page 498
9.4 Singularities of matrices......Page 515
10 Saks spaces and bounded domains......Page 530
10.1 Saks spaces and mixed topologies......Page 531
10.2 The Saks space m fl WE......Page 537
10.3 A theorem of Mazur-Orlicz type......Page 542
10.4 b-comparison through quotient representations......Page 544
10.5 Notes on Chapter 10......Page 551
11.1 An inclusion theorem......Page 553
11.2 Gliding hump and oscillating properties......Page 555
11.3 Theorems of Toeplitz-Silverman type via sectional convergence and .........Page 561
11.4 Barrelled K-spaces......Page 565
11.5 The sequences of zeros and ones in a sequence space......Page 572
11.6 Notes on Chapters 9 and 11......Page 576
Bibliography......Page 578
Index......Page 590