Kannappan Functional Equations and Inequalities with Applications

1. Auflage 2009
ISBN: 978-0-387-89492-8
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 810 Seiten

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 entrée into an active area of mathematical exploration for engineers and other scientists who would benefit from this careful, rigorous exposition.

Autoren/Hrsg.

Kannappan, Palaniappan

Inhaltsverzeichnis

1;Preface;6
2;Contents;14
3;1 Basic Equations: Cauchy and Pexider Equations;23
3.1;1.1 Additive Equations;24
3.2;1.2 Algebraic Conditions—Derivation;29
3.3;1.3 Additive Equation on C, Rn, and R+;37
3.4;1.4 Alternative Equations, Restricted Domains, and Conditional Cauchy Equations;42
3.5;1.5 The Other Cauchy Equations;48
3.6;1.6 Some Generalizations of the Cauchy Equations;56
3.7;1.7 Extensions;75
3.8;1.8 Applications;90
4;2 Matrix Equations;106
4.1;2.1 Multiplicative Equation;112
4.2;2.2 Cosine Matrix Equation;119
5;3 Trigonometric Functional Equations;125
5.1;3.1 Mixed Trigonometric Equations;127
5.2;3.2 Cosine Equation on Number Systems;133
5.3;3.3 (C) on Groups and Vector Spaces;140
5.4;3.4 Solution of (CE) on a Non-Abelian Group;144
5.5;3.5 Survey—Summary of Stetkaer;229
6;4 Quadratic Functional Equations;240
6.1;4.1 General Solution and Properties;240
6.2;4.2 General Solution on a Complex Linear Space—Sesquilinear Solution;244
6.3;4.3 Regular Solutions;247
6.4;4.4 Generalizations and Equivalent Forms of (Q);249
6.5;4.5 Equivalence to (Q);251
6.6;4.6 Generalizations;253
6.7;4.7 More Generalizations;257
6.8;4.8 Another Form of Quadratic Function;259
6.9;4.9 Entire Functions and Quadratic Equations;260
6.10;4.10 Summary by Stetkaer;263
7;5 Characterization of Inner Product Spaces;265
7.1;5.1 Frechet’s Equation;267
7.2;5.2 Geometric Characterization;293
8;6 Stability;313
8.1;6.1 Stability of the Additive Equation;314
8.2;6.2 Stability—Multiplicative Equations;320
8.3;6.3 Stability—Logarithmic Function;321
8.4;6.4 Stability—Trigonometric Functions;323
8.5;6.5 Stability of the Equation f (x + y) + g(x y) = h(x)k( y);336
8.6;6.6 Stability of the Sine Functional Equation;337
8.7;6.7 Stability—Alternative Cauchy Equation;337
8.8;6.8 Stability—Wave Equation;339
8.9;6.9 Stability—Polynomial Equation;340
8.10;6.10 Stability—Quadratic Equation;341
9;7 Characterization of Polynomials;346
9.1;7.1 Polynomials;346
9.2;7.2 More Characterizations of Polynomials of Degree Two;349
9.3;7.3 Generalization;353
9.4;7.4 Another Generalization—Divided Difference;354
9.5;7.5 Generalization of Divided Difference;357
9.6;7.6 Problem ofW. Rudin and a Generalization;357
9.7;7.7 Generalization of Rudin’s Problem;359
9.8;7.8 Frechet’s Result;360
9.9;7.9 Polynomials in Several Variables;362
9.10;7.10 Quadratic Polynomials in Two Variables;363
9.11;7.11 Functional Equations on Groups;364
9.12;7.12 Rudin’s Problem on Groups;366
9.13;7.13 Generalization;369
10;8 Nondifferentiable Functions;375
10.1;8.1 Weierstrass Functions;375
10.2;8.2 Wünderlich’s Function;377
10.3;8.3 Takagi Functions;377
10.4;8.4 van der Waerden Type Function;378
10.5;8.5 Functional Equation Characterization of w;380
10.6;8.6 Nondifferentiability of w;381
10.7;8.7 Riemann’s Function;383
10.8;8.8 Knopp Functions;384
10.9;8.9 Generalization;385
11;9 Characterization of Groups, Loops, and Closure Conditions;387
11.1;9.1 Notation and Definitions;387
11.2;9.2 Closure Conditions;388
11.3;9.3 Groups;389
11.4;9.4 Abelian Groups;390
11.5;9.5 Functional Equation of Identities;397
11.6;9.6 Functional Equations Arising Out of Bol, Moufang, and Extra Equations;400
11.7;9.7 GD-groupoid;408
11.8;9.8 More Identities;408
11.9;9.9 Left Inverse Property (l.i.p.), Crossed-Inverse (c.i.), and Weak Inverse Property ( w. i. p.) Loops;410
11.10;9.10 Steiner Loops;411
11.11;9.11 Bol Loop and Power Associativity;414
11.12;9.12 More Functional Equations;414
11.13;9.13 Generalized Groupoids;416
12;10 Functional Equations from Information Theory;418
12.1;10.1 Introduction;418
12.2;10.2 Notation, Basic Notions, and Preliminaries;420
12.2.1;10.2a Fundamental Equation of Information—Axiomatic Characterizations;425
12.2.2;10.2b The Fundamental Equation of Information Theory;427
12.2.3;10.2c Some Generalizations of (FEI);433
12.2.4;10.2d Sum Form Functional Equation (SFE) and Its Generalizations;439
12.2.5;10.2d.1 Representation and Characterization;439
12.2.6;10.2e Other Measures of Information—Entropy of Type;447
12.2.7;10.2f Directed Divergence (dd) and Inaccuracy (KI);448
12.2.8;10.2f.1 Generalized Directed Divergence;452
12.2.9;10.2g Sum Form Distance Measures;456
12.2.10;10.2h Kullback-Leibler Type Distance Measures;459
12.2.11;10.2i Symmetric Distance Measures;460
12.2.12;10.2j Some Functional Equations;462
12.2.13;10.2k Weighted Entropies;463
12.3;10.3 Mixed Theory of Information—Inset Measures;467
12.4;10.4 Applications;477
13;11 Abel Equations and Generalizations;483
13.1;11.1 Solutions of Abel Equations;484
13.2;11.2 Generalizations and Information Measures;499
14;12 Regularity Conditions— Christensen Measurability;507
14.1;12.1 Some General Results;511
14.2;12.2 Applications;514
14.3;12.3 Christensen Measurability;517
14.4;12.4 Functional Equations (Characterizing) from Trigonometric Functions;519
15;13 Difference Equations;524
15.1;13.1 Cauchy Difference;524
15.2;13.2 Quadratic Differences;539
15.3;13.3 Pexider Difference;543
16;14 Characterization of Special Functions;549
16.1;14.1 Gamma Function;549
16.2;14.2 Beta Function;561
16.3;14.3 Riemann’s Zeta Function;567
16.4;14.4 Singular Functions;569
17;15 Miscellaneous Equations;574
17.1;15.1 A General Method: Method of Determinants;574
17.2;15.2 Means;581
17.3;15.3 Some Comments about the Logarithmic Function;592
17.4;15.4 D’Alembert’s Equation Revisited;599
17.5;15.5 Polynomials Revisited;608
17.6;15.6 Inner Products Revisited;611
18;16 General Inequalities;617
18.1;16.1 Cauchy Functional Inequalities;617
18.2;16.2 Subadditive and Superadditive Functions;620
18.3;16.3 Logarithmic Inequality;622
18.4;16.4 Multiplicative Inequality;624
18.5;16.5 Convexity;624
18.6;16.6 Trigonometric Functional Inequality;627
18.7;16.7 Cosine and Sine Functional Inequalities;634
18.8;16.8 Functional Equation Concerning the Parallelogram Identity— Quadratic Inequality;636
18.9;16.9 Inequalities for the Gamma and Beta Functions via some Classical Results;637
18.10;16.10 Simpson’s Inequality and Applications;644
18.11;16.11 Applications for Special Means;646
18.12;16.12 Inequalities from Information Theory;648
18.13;16.13 Reproducing Scoring Systems;661
18.14;16.14 More Inequalities from Information Theory;666
18.15;16.15 Inequalities from Inner Product Spaces;668
18.16;16.16 Miscellaneous Inequalities;671
19;17 Applications;678
19.1;17.1 Binomial Expansion;678
19.2;17.2 Scalar or Dot Product;680
19.3;17.3 Economics;682
19.4;17.4 Business Mathematics—Interest;689
19.5;17.5 Physics;691
19.6;17.6 Topology;703
19.7;17.7 Digital Filtering;719
19.8;17.8 Geometry;720
19.9;17.9 Field Homomorphisms;722
19.10;17.10 Pythagorean Functional Equation;724
19.11;17.11 Statistics;726
19.12;17.12 Information Theory;743
19.13;17.13 Behavioural Sciences;754
20;Symbols;766
21;Bibliography;768
22;Author Index;803
23;Subject Index;810