Functional Equations and Inequalities with Applications (eBook)

Palaniappan Kannappan (Autor)

eBook Download: PDF

2009 | 1. Auflage
XXIII, 816 Seiten
Springer US (Verlag)
978-0-387-89492-8 (ISBN)

Infos

Functional Equations and Inequalities with Applications presents a comprehensive, nearly encyclopedic, study of the classical topic of functional equations.  This self-contained monograph explores all aspects of functional equations and their applications to related topics,  such as differential equations, integral equations, the Laplace transformation, the calculus of finite differences, and many other basic tools in analysis. Each chapter examines a particular family of equations and gives an in-depth study of its applications as well as examples and exercises to support the material.

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Functional Equations and Inequalities with Applications presents a comprehensive, nearly encyclopedic, study of the classical topic of functional equations. Nowadays, the field of functional equations is an ever-growing branch of mathematics with far-reaching applications; it is increasingly used to investigate problems in mathematical analysis, combinatorics, biology, information theory, statistics, physics, the behavioral sciences, and engineering.This self-contained monograph explores all aspects of functional equations and their applications to related topics, such as differential equations, integral equations, the Laplace transformation, the calculus of finite differences, and many other basic tools in analysis. Each chapter examines a particular family of equations and gives an in-depth study of its applications as well as examples and exercises to support the material.The book is intended as a reference tool for any student, professional (researcher), or mathematician studying in a field where functional equations can be applied. It can also be used as a primary text in a classroom setting or for self-study. Finally, it could be an inspiring entree into an active area of mathematical exploration for engineers and other scientists who would benefit from this careful, rigorous exposition.

Inhalt

Preface 6
Contents 14
1 Basic Equations: Cauchy and Pexider Equations 23
1.1 Additive Equations 24
1.2 Algebraic Conditions—Derivation 29
1.3 Additive Equation on C, Rn, and R+ 37
1.4 Alternative Equations, Restricted Domains, and Conditional Cauchy Equations 42
1.5 The Other Cauchy Equations 48
1.6 Some Generalizations of the Cauchy Equations 56
1.7 Extensions 75
1.8 Applications 90
2 Matrix Equations 106
2.1 Multiplicative Equation 112
2.2 Cosine Matrix Equation 119
3 Trigonometric Functional Equations 125
3.1 Mixed Trigonometric Equations 127
3.2 Cosine Equation on Number Systems 133
3.3 (C) on Groups and Vector Spaces 140
3.4 Solution of (CE) on a Non-Abelian Group 144
3.5 Survey—Summary of Stetkaer 229
4 Quadratic Functional Equations 240
4.1 General Solution and Properties 240
4.2 General Solution on a Complex Linear Space—Sesquilinear Solution 244
4.3 Regular Solutions 247
4.4 Generalizations and Equivalent Forms of (Q) 249
4.5 Equivalence to (Q) 251
4.6 Generalizations 253
4.7 More Generalizations 257
4.8 Another Form of Quadratic Function 259
4.9 Entire Functions and Quadratic Equations 260
4.10 Summary by Stetkaer 263
5 Characterization of Inner Product Spaces 265
5.1 Frechet’s Equation 267
5.2 Geometric Characterization 293
6 Stability 313
6.1 Stability of the Additive Equation 314
6.2 Stability—Multiplicative Equations 320
6.3 Stability—Logarithmic Function 321
6.4 Stability—Trigonometric Functions 323
6.5 Stability of the Equation f (x + y) + g(x y) = h(x)k( y) 336
6.6 Stability of the Sine Functional Equation 337
6.7 Stability—Alternative Cauchy Equation 337
6.8 Stability—Wave Equation 339
6.9 Stability—Polynomial Equation 340
6.10 Stability—Quadratic Equation 341
7 Characterization of Polynomials 346
7.1 Polynomials 346
7.2 More Characterizations of Polynomials of Degree Two 349
7.3 Generalization 353
7.4 Another Generalization—Divided Difference 354
7.5 Generalization of Divided Difference 357
7.6 Problem ofW. Rudin and a Generalization 357
7.7 Generalization of Rudin’s Problem 359
7.8 Frechet’s Result 360
7.9 Polynomials in Several Variables 362
7.10 Quadratic Polynomials in Two Variables 363
7.11 Functional Equations on Groups 364
7.12 Rudin’s Problem on Groups 366
7.13 Generalization 369
8 Nondifferentiable Functions 375
8.1 Weierstrass Functions 375
8.2 Wünderlich’s Function 377
8.3 Takagi Functions 377
8.4 van der Waerden Type Function 378
8.5 Functional Equation Characterization of w 380
8.6 Nondifferentiability of w 381
8.7 Riemann’s Function 383
8.8 Knopp Functions 384
8.9 Generalization 385
9 Characterization of Groups, Loops, and Closure Conditions 387
9.1 Notation and Definitions 387
9.2 Closure Conditions 388
9.3 Groups 389
9.4 Abelian Groups 390
9.5 Functional Equation of Identities 397
9.6 Functional Equations Arising Out of Bol, Moufang, and Extra Equations 400
9.7 GD-groupoid 408
9.8 More Identities 408
9.9 Left Inverse Property (l.i.p.), Crossed-Inverse (c.i.), and Weak Inverse Property ( w. i. p.) Loops 410
9.10 Steiner Loops 411
9.11 Bol Loop and Power Associativity 414
9.12 More Functional Equations 414
9.13 Generalized Groupoids 416
10 Functional Equations from Information Theory 418
10.1 Introduction 418
10.2 Notation, Basic Notions, and Preliminaries 420
10.2a Fundamental Equation of Information—Axiomatic Characterizations 425
10.2b The Fundamental Equation of Information Theory 427
10.2c Some Generalizations of (FEI) 433
10.2d Sum Form Functional Equation (SFE) and Its Generalizations 439
10.2d.1 Representation and Characterization 439
10.2e Other Measures of Information—Entropy of Type 447
10.2f Directed Divergence (dd) and Inaccuracy (KI) 448
10.2f.1 Generalized Directed Divergence 452
10.2g Sum Form Distance Measures 456
10.2h Kullback-Leibler Type Distance Measures 459
10.2i Symmetric Distance Measures 460
10.2j Some Functional Equations 462
10.2k Weighted Entropies 463
10.3 Mixed Theory of Information—Inset Measures 467
10.4 Applications 477
11 Abel Equations and Generalizations 483
11.1 Solutions of Abel Equations 484
11.2 Generalizations and Information Measures 499
12 Regularity Conditions— Christensen Measurability 507
12.1 Some General Results 511
12.2 Applications 514
12.3 Christensen Measurability 517
12.4 Functional Equations (Characterizing) from Trigonometric Functions 519
13 Difference Equations 524
13.1 Cauchy Difference 524
13.2 Quadratic Differences 539
13.3 Pexider Difference 543
14 Characterization of Special Functions 549
14.1 Gamma Function 549
14.2 Beta Function 561
14.3 Riemann’s Zeta Function 567
14.4 Singular Functions 569
15 Miscellaneous Equations 574
15.1 A General Method: Method of Determinants 574
15.2 Means 581
15.3 Some Comments about the Logarithmic Function 592
15.4 D’Alembert’s Equation Revisited 599
15.5 Polynomials Revisited 608
15.6 Inner Products Revisited 611
16 General Inequalities 617
16.1 Cauchy Functional Inequalities 617
16.2 Subadditive and Superadditive Functions 620
16.3 Logarithmic Inequality 622
16.4 Multiplicative Inequality 624
16.5 Convexity 624
16.6 Trigonometric Functional Inequality 627
16.7 Cosine and Sine Functional Inequalities 634
16.8 Functional Equation Concerning the Parallelogram Identity— Quadratic Inequality 636
16.9 Inequalities for the Gamma and Beta Functions via some Classical Results 637
16.10 Simpson’s Inequality and Applications 644
16.11 Applications for Special Means 646
16.12 Inequalities from Information Theory 648
16.13 Reproducing Scoring Systems 661
16.14 More Inequalities from Information Theory 666
16.15 Inequalities from Inner Product Spaces 668
16.16 Miscellaneous Inequalities 671
17 Applications 678
17.1 Binomial Expansion 678
17.2 Scalar or Dot Product 680
17.3 Economics 682
17.4 Business Mathematics—Interest 689
17.5 Physics 691
17.6 Topology 703
17.7 Digital Filtering 719
17.8 Geometry 720
17.9 Field Homomorphisms 722
17.10 Pythagorean Functional Equation 724
17.11 Statistics 726
17.12 Information Theory 743
17.13 Behavioural Sciences 754
Symbols 766
Bibliography 768
Author Index 803
Subject Index 810

Produkt-Details

• Erscheint lt. Verlag: 10.6.2009
• Reihe/Serie: Springer Monographs in Mathematics
• Zusatzinfo: XXIV, 816 p. 16 illus.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik
• Schlagworte: Cauchy equations • Difference Equations • Information Theory • information thoery • matrix equations • nondifferentiable functions • quadratic functional equations • Special Functions
• ISBN-10: 0-387-89492-6 / 0387894926
• ISBN-13: 978-0-387-89492-8 / 9780387894928