Advanced topics in mathematical analysis

Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Table of Contents......Page 6
Preface......Page 14
Contributors......Page 20
 1.1 Introduction, history, and motivation......Page 22
 1.2 RKHSs in measurable category: new settings for positivity conditions......Page 24
 1.3 L2loc(λ) vs L2(λ): Local versus global L2-conditions......Page 28
 1.4 Continuous networks: Atomic versus non-atomic models......Page 31
 1.5 New correspondences......Page 37
 A.1 Applications of RKHSs: An overview......Page 45
 A.2 Duality of representations, and some of their applications......Page 53
 2.1 Matrix monotone function......Page 60
 2.2 Matrix means......Page 67
 2.3 Matrix power mean......Page 72
 2.4 Karcher mean......Page 79
 3.1 Introduction......Page 92
 3.2 Semidirect products......Page 93
 3.3 Linear groups......Page 95
 3.4 Affine groups......Page 98
 3.5 Invariant functions and measures......Page 101
 3.6 Spherical functions and functional equations......Page 104
 3.7 Functional equations on Euclidean motion groups......Page 108
 3.8 Affine groups of generalized orthogonal groups......Page 111
 4.1 Brief introduction......Page 116
  4.1.1 Linear spaces......Page 117
  4.1.2 Topological spaces......Page 119
 4.2 Topological linear spaces......Page 120
 4.3 Topology of a linear space and an algebra, defined by a collection of F-seminorms......Page 124
 4.4 Locally pseudoconvex spaces and algebras......Page 130
  4.4.1 Saturation of a collection of seminorms......Page 135
 4.5 Locally bounded space and algebra......Page 137
 4.6 Main types of locally pseudoconvex algebras......Page 139
 4.7 Generalizations of Gelfand-Mazur theorem for locally pseudoconvex division algebras......Page 144
 4.8 Generalizations of Arens-Michael theorem for locally pseudoconvex algebras......Page 152
 4.9 Generalization of Akkar teorem for locally pseudoconvex algebras......Page 157
 5.1 Introduction......Page 168
 5.3 VMO class......Page 169
 5.4 Morrey-Campanato spaces......Page 170
 5.5 Generalized Morrey spaces......Page 173
 5.6 Mixed Morrey spaces......Page 176
   5.7.1.1 Preliminary tools......Page 177
   5.7.1.2 Nondivergence form elliptic equations......Page 180
   5.7.1.3 Representation formula......Page 181
   5.7.1.4 Regularity......Page 182
   5.7.1.5 Divergence form elliptic equations......Page 187
   5.7.2.1 Preliminary tools......Page 193
   5.7.2.2 Nondivergence form parabolic equations......Page 198
   5.7.2.3 Divergence form parabolic equations......Page 203
   5.7.3.1 Preliminary tools......Page 204
   5.7.3.2 Nondivergence form ultraparabolic equations......Page 206
   5.7.3.3 Divergence form ultraparabolic equations......Page 209
 6.1 Introduction......Page 214
 6.2 Some composite functionals......Page 215
 6.3 Applications for Jensen\'s inequality......Page 219
 6.4 Applications for Hölder\'s inequality......Page 222
 6.5 Applications for Minkowski\'s inequality......Page 225
 6.6 Applications for Cauchy-Bunyakovsky-Schwarz\'s inequality......Page 227
 6.7 Applications for Čebyšev\'s inequality......Page 231
 6.8 Applications for Hermite-Hadamard inequalities......Page 232
 6.9 Applications for convex functions......Page 235
 7.1 Introduction......Page 240
  7.2.1 Weighted statistical convergence......Page 242
  7.2.2 Difference operator of order m based on (p, q)-integers......Page 244
 7.3 Some new definitions involving generalized double difference operator......Page 246
 7.4 Some inclusion relations......Page 252
 7.5 Korovkin type approximation theorem......Page 255
 8.1 Introduction......Page 266
  8.1.1 Notations and terminologies......Page 267
  8.1.2 A brief motivation and history......Page 269
 8.2 Relation between Birkhoff-James orthogonality in B(X) and X......Page 270
 8.3 Characterization of IPS in terms of operator norm attainment......Page 273
 8.4 Operator orthogonality on finite-dimensional Banach spaces......Page 274
 8.5 Operator orthogonality on infinite-dimensional normed spaces......Page 284
 8.6 Operator orthogonality on Hilbert spaces......Page 288
 8.7 Operator orthogonality and smoothness of bounded linear operators......Page 290
 8.8 Characterization of operator orthogonality for finite-dimensional spaces......Page 294
 8.9 Symmetry of operator orthogonality......Page 296
 9.1 Fixed point theory with graph structure......Page 306
 9.2 Fixed points of graphic contraction mappings......Page 310
 9.3 Weakly commuting multivalued mappings endowed with directed graph......Page 317
 9.4 Coincidence and common fixed points......Page 318
10: Some Subclasses of Analytic Functions and Their Properties......Page 330
 10.1 Introduction......Page 331
 10.2 Conditions for subclasses S* C(α, β; γ) and TS* C(α, β; γ)......Page 333
 10.3 Conditions for analytic functions involving gamma function......Page 339
 10.4 Integral operators of functions Fλ,μ(z) and Gλ,μ(z)......Page 341
 10.5 The various properties of certain subclasses of analytic functions of complex order......Page 344
  10.5.1 Order of starlikeness and radius of convexity for classes TC(α; τ ) and TS* (α; τ )......Page 352
 10.6 Upper bound for second Hankel determinant of certain sub-class of analytic and bi-univalent functions......Page 354
  10.6.1 Upper bound estimate for the second Hankel determinant of the class QΣ (G, β, t)......Page 358
 10.7 Fekete-Szegö problem for certain subclass of analytic and bi-univalent functions......Page 364
  10.7.1 Fekete-Szegö problem for class MΣ (ϕ, β) ......Page 367
  10.7.2 Concluding remarks......Page 371
 10.8 Some starlikeness and convexity properties for two new p−valent integral operators......Page 372
 10.9 Convexity of integral operators Fδ,λ,μ n,p,l (z) and Gδ,λ,μ n,p,l (z)......Page 376
 10.10 Starlikeness of integral operators Fδ,λ,μ n,p,l (z) and Gδ,λ,μ n,p,l (z)......Page 381
 11.1 Introduction......Page 392
  11.1.1 Motivation and background......Page 393
 11.2 Preliminaries......Page 394
 11.3 On Banach *-algebras induced by orthogonal projections......Page 395
 11.4 Weighted-semicircular elements induced by Q......Page 398
 11.5 Semicircular elements induced by Q......Page 403
 11.6 The free product Banach *-probability space ⋆j∈Z  LQ(j)......Page 404
 11.7 Weighted-semicircular processes on free lterizations......Page 407
 11.8 Free stochastic integrals for Xj......Page 411
 11.9 Glossary......Page 422
 12.1 Introduction......Page 424
  12.1.1 Preliminaries and definitions......Page 425
 12.2 Fractional difference operator......Page 429
 12.3 Related sequence spaces......Page 433
 12.4 Application to fractional derivatives......Page 435
 12.5 Geometrical interpretation......Page 440
 13.1 Introduction......Page 446
 13.2 Some identities......Page 449
 13.3 Inequalities for bounded function......Page 457
 13.4 Composite inequalities for functions of bounded variation......Page 462
 14.1 Introduction......Page 470
  14.1.1 Preliminaries and definitions......Page 471
 14.2 Main results......Page 474
 14.3 Possible model matrices of kn – 2 matrix......Page 476
  14.4.2 New method for modeling quadratic functional equations......Page 478
 14.5 Numerical example......Page 479
 14.6 Modeling the functional equations corresponding to logical matrix......Page 484
 14.7 Conclusion......Page 486
  14.7.1 Scope for further study......Page 487
 15.1 Introduction and preliminaries......Page 490
  15.1.1 Preliminaries......Page 496
 15.2 A renorming of c0 and a large class of non-weakly compact, closed, bounded, and convex subsets with FPP for affine non-expansive mappings......Page 497
  15.2.1 A large class of non-weakly compact, c.b.c. sets in c0 having FPP for affine ║ · ║- nonexpansive  mappings......Page 500
 15.3 Another renorming of c0 allowing larger class of non-weakly compact, closed, bounded, and convex subsets to have FPP for affine nonexpansive mappings......Page 517
 15.4 c0 can be renormed to have the fixed point property for affine nonexpansive mappings......Page 524
 15.5 Family of equivalent norms on c0 with FPP for affine nonexpansive mappings......Page 539
 16.1 Introduction......Page 550
 16.2 Basic concepts and definitions in summability theory......Page 551
 16.3 A Steinhaus type theorem involving the sequence spaces Λr, r ≥ 1 being integer......Page 554
 16.4 On theorems of Steinhaus type proved by Maddox......Page 557
 16.5 On a theorem of Steinhaus type of Fridy over valued fields ......Page 561
 16.6 Some Steinhaus type theorems of Natarajan over valued fields......Page 567
 16.7 Some more Steinhaus type theorems over valued fields I......Page 571
 16.8 Some more Steinhaus type theorems over valued fields II......Page 576
17: The Interplay Between Topological Algebras Theory and Algebras of Holomorphic Functions......Page 584
 17.2 Topological algebras......Page 585
   17.2.1.1 Banach algebras: characters and spectrum......Page 586
   17.2.1.4 Banach algebras: analytical properties of the spectrum......Page 587
   17.2.1.5 C*-algebras......Page 590
   17.2.2.1 Fréchet algebras: characters and spectrum......Page 591
   17.2.2.2 Fréchet algebras: Michael\'s problem......Page 592
  17.2.3 LB algebras and LF algebras......Page 593
 17.3 Algebras of holomorphic functions......Page 594
   17.3.1.1 Stein spaces and Stein manifolds......Page 595
   17.3.1.3 The envelope of holomorphy problem......Page 596
  17.3.2 The algebra of bounded holomorphic functions H∞ (D)......Page 599
  17.3.3 The algebras of holomorphic functions continuous (or more) up to the boundary Ak(D)......Page 601
   17.3.3.1 Šilov boundary of Ak(D)......Page 602
   17.3.3.2 Finitely generated maximal ideals......Page 603
  17.3.5 The algebra of holomorphic functions with polynomial growth P(D)......Page 604