Functional equations and inequalities with applications / by Pl. Kannappan.

Format:

Book

Author/Creator:

Kannappan, Pl. (Palaniappan)

Series:

Springer monographs in mathematics

Language:

English

Subjects (All):

Functional equations.

Inequalities (Mathematics).

Physical Description:

xxiii, 810 pages ; 24 cm.

Place of Publication:

New York ; London : Springer, 2009.

Summary:

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

Contents:

1 Basic Equations: Cauchy and Pexider Equations 1

1.1 Additive Equations 2

1.1.1 Discontinuous Solutions 6

1.2 Algebraic Conditions-Derivation 7

1.2.1 More Algebraic Conditions 11

1.3 Additive Equation on C, Rn, and R+ 15

1.3.1 Additive Equation on Complex Numbers 15

1.3.2 More Algebraic Conditions 17

1.4 Alternative Equations, Restricted Domains, and Conditional Cauchy Equations 20

1.4.1 Alternative Equations 21

1.4.2 Restricted Domains and Conditional Cauchy Equations 23

1.4.3 Mikusiński's Functional Equation 24

1.5 The Other Cauchy Equations 26

1.5.1 Exponential Equation 27

1.5.2 Logarithmic Equation 29

1.5.3 Characterization of Exponential and Logarithmic Functions 30

1.5.4 Multiplicative Equation 33

1.6 Some Generalizations of the Cauchy Equations 34

1.6.1 Jensen's Equation 34

1.6.2 Pexider's Equations 39

1.6.3 Some Generalizations 43

1.6.4 Some Special Cases 45

1.6.5 More Generalizations 52

1.7 Extensions 53

1.7.1 Extension of the Additive Function on [0, 1] 56

1.7.2 Quasiextensions 59

1.7.3 Extension of the Pexider Equation 60

1.7.4 Extension of the Logarithmic Equation 64

1.7.5 Exponential Extension 65

1.7.6 Multiplicative Extension 66

1.7.7 Extension of Derivations 67

1.7.8 Almost Everywhere Extension 67

1.8 Applications 68

1.8.1 Economics 68

1.8.2 Area 72

1.8.3 Allocation Problem: Characterization of Weighted Arithmetic Means 74

1.8.4 Sum of Powers of First n Natural Numbers and Sum of Powers on Arithmetic Progressions 76

1.8.5 More Sums Using the Additive Cauchy Equation 81

1.8.6 Application in Combinatorics and Genetics 82

2 Matrix Equations 85

2.1 Multiplicative Equation 91

2.2 Cosine Matrix Equation 98

3 Trigonometric Functional Equations 105

3.1 Mixed Trigonometric Equations 107

3.2 Cosine Equation on Number Systems 113

3.3 (C) on Groups and Vector Spaces 120

3.4 Solution of (CE) on a Non-Abelian Group 124

3.4.1 Discussion 131

3.4.2 (C) on Abstract Spaces 133

3.4.3 Jacobi's Elliptic Function Solution 139

3.4.4 More Characterizations in a Single Variable 145

3.4.5 More on Sine Functions on a Vector Space 158

3.4.6 Sine Solution 170

3.4.7 Characterization of the Sine 172

3.4.8 General Trigonometric Functional Equations-The Addition and Subtraction Formulas 174

3.4.9 Vibration of String Equation (VS) 181

3.4.10 Wilson's Equations 186

3.4.11 Analytic Solutions 187

3.4.12 Equation (3.72) on Analytic Functions 192

3.4.13 Some Generalizations 197

3.4.14 Levi-Civitá Functional Equation, Convolution Type Functional Equations, and Generalization of Cauchy-Pexider Type and d'Alembert Equations 202

3.4.15 Operator-Valued Solution of (C) 203

3.4.16 Solution of Equation (3.111) 204

3.4.17 Inner Product Version of (3.109) 205

3.4.18 A Functional Equation of d'Alembert's Type 207

3.5 Survey-Summary of Stetkaer 209

3.5.1 Abstract 210

3.5.2 The Abelian Solution and Related Extensions of It 210

3.5.3 d'Alembert's Functional Equation 215

3.5.4 Wilson's Functional Equation 216

3.5.5 A Variant of Wilson's Functional Equation 217

3.5.6 Other Equations 218

4 Quadratic Functional Equations 221

4.1 General Solution and Properties 221

4.2 General Solution on a Complex Linear Space-Sesquilinear Solution 225

4.3 Regular Solutions 228

4.4 Generalizations and Equivalent Forms of (Q) 230

4.5 Equivalence to (Q) 232

4.6 Generalizations 234

4.7 More Generalizations 238

4.8 Another Form of Quadratic Function 240

4.9 Entire Functions and Quadratic Equations 241

4.10 Summary by Stetkaer [750] 244

4.10.1 The Quadratic Functional Equation 244

5 Characterization of Inner Product Spaces 247

5.1 Frčhet's Equation 249

5.1.1 The Parallelepiped Law 251

5.1.2 Parallelogram Identity 253

5.1.3 More on Frčhet's Result 254

5.1.4 More Characterizations 256

5.1.5 Some Generalizations 263

5.1.5.1 Solution of a Generalization of the Frčhet Equation 263

5.1.6 Some More Characterizations 266

5.2 Geometric Characterization 275

5.2.1 Some Generalizations 277

5.2.2 Solution of an Equation Related to Ptolemaic Inequality 283

5.2.3 Orthogonal Additivity and I.P.S. 285

5.2.4 Diminnie Orthogonality 285

6 Stability 295

6.1 Stability of the Additive Equation 296

6.2 Stability-Multiplicative Equations 302

6.3 Stability-Logarithmic Function 303

6.4 Stability-Trigonometric Functions 305

6.4.1 Stability for Vector-Valued Functions 315

6.5 Stability of the Equation f(x+y)+g(x-y)=h(x)k(y) 318

6.6 Stability of the Sine Functional Equation 319

6.7 Stability-Alternative Cauchy Equation 319

6.8 Stability-Wave Equation 321

6.9 Stability-Polynomial Equation 322

6.10 Stability-Quadratic Equation 323

7 Characterization of Polynomials 329

7.1 Polynomials 329

7.2 More Characterizations of Polynomials of Degree Two 332

7.3 Generalization 336

7.3.1 First Generalization Using Derivatives 336

7.3.2 Second Generalization Without Using Any Regularity Condition 337

7.4 Another Generalization-Divided Difference 337

7.5 Generalization of Divided Difference 340

7.6 Problem of W. Rudin and a Generalization 340

7.7 Generalization of Rudin's Problem 342

7.8 Frčhet's Result 343

7.9 Polynomials in Several Variables 345

7.10 Quadratic Polynomials in Two Variables 346

7.11 Functional Equations on Groups 347

7.12 Rudin's Problem on Groups 349

7.13 Generalization 352

8 Nondifferentiable Functions 359

8.1 Weierstrass Functions 359

8.2 Wünderlich's Function 361

8.3 Takagi Functions 361

8.4 van der Waerden Type Function 362

8.5 Functional Equation Characterization of ω 364

8.6 Nondifferentiability of ω 365

8.7 Riemann's Function 367

8.8 Knopp Functions 368

8.9 Generalization 369

9 Characterization of Groups, Loops, and Closure Conditions 371

9.1 Notation and Definitions 371

9.2 Closure Conditions 372

9.3 Groups 373

9.4 Abelian Groups 374

9.5 Functional Equation of Identities 381

9.6 Functional Equations Arising Out of Bol, Moufang, and Extra Equations 384

9.6.1 Bol Equation 384

9.6.2 Moufang and Extra Loops 390

9.6.3 Extra Loop 391

9.6.4 Characterizations 391

9.6.5 Characterization of Moufang Loops 391

9.6.6 Extra Loops 391

9.6.7 Groups 392

9.7 GD-groupoid 392

9.8 More Identities 392

9.8.1 Entropic, Bisymmetric, or Mediality Identity 392

9.9 Left Inverse Property (l.i.p.), Crossed-Inverse (c.i.), and Weak Inverse Property (w.i.p.) Loops 394

9.9.1 l.i.p. Loops 394

9.9.2 c.i. Loops 395

9.9.3 w.i.p.

Loops 395

9.10 Steiner Loops 395

9.11 Bol Loop and Power Associativity 398

9.12 More Functional Equations 398

9.13 Generalized Groupoids 400

9.13.1 Generalized Associativity 400

9.13.2 Generalized Bisymmetry 400

10 Functional Equations from Information Theory 403

10.1 Introduction 403

10.2 Notation, Basic Notions, and Preliminaries 405

10.2.1 Properties, Postulates, and Axioms 406

10.2.2 Desirable Properties-Postulates 406

10.2.3 Characterization of Information Measures 409

10.2.4 Shannon Entropy and Some of Its Generalizations 410

10.2a Fundamental Equation of Information-Axiomatic Characterizations 410

10.2b The Fundamental Equation of Information Theory 412

10.2c Some Generalizations of (FEI) 418

10.2.5 General Solution of Equation (10.21) 420

10.2d Sum Form Functional Equation (SFE) and Its Generalizations 424

10.2d.1 Representation and Characterization 424

10.2e Other Measures of Information-Entropy of Type β, Hβn 432

10.2f Directed Divergence (dd) and Inaccuracy (KI) 433

10.2f.1 Generalized Directed Divergence 437

10.2g Sum Form Distance Measures 441

10.2h Kullback-Leibler Type Distance Measures 444

10.2i Symmetric Distance Measures 445

10.2j Some Functional Equations 447

10.2k Weighted Entropies 448

10.3 Mixed Theory of Information-Inset Measures 452

10.3.1 Characterizations 454

10.3.1.1 Characterization of Inset Deviation Entropies 456

10.4 Applications 462

10.4.1 Continuous Shannon Measure and Shannon-Wiener Inset Information 462

10.4.2 Theory of Gambles 463

10.4.3 Recursive Inset Entropies of Multiplicative Type 465

11 Abel Equations and Generalizations 469

11.1 Solutions of Abel Equations 470

11.1.1 (AFE1)-Exponential Equation f(z+w)=f(z)f(w) 470

11.1.2 (AFE2)-Interation Equation f((̜x))=f(x)+1 472

11.1.3 (AFE3)-Associative, Commutative Equations 472

11.1.4 (AFE4)-Arctan Equation 473

11.1.5 (AFE5)-Trig Equation (̜x+y)=(̜x)f(y)+(̜y)f(x) 473

11.1.6 (AFE6)-(̜x+y)=g(xy)+h(x-y) 473

11.1.7 (AFE7)-(̜x)+(̜y)=(̜xf(y)+yf(x)) 479

11.1.8 System of Equations (AFE8) and (AFE8a) 480

11.1.9 (AFE9) and (AFE9a) 482

11.2 Generalizations and Information Measures 485

12 Regularity Conditions-Christensen Measurability 493

12.1 Some General Results 497

12.2 Applications 500

12.3 Christensen Measurability 503

12.4 Functional Equations (Characterizing) from Trigonometric Functions 505

13 Difference Equations 511

13.1 Cauchy Difference 511

13.1.1 Differences that Depend on the Product 516

13.1.2 Pompeiu Functional Equation and Its Generalizations 523

13.1.3 Solution of the Functional Equation (13.24a) 525

13.1.4 Solution of the Functional Equation (13.24b) 526

13.2 Quadratic Differences 526

13.2.1 Differences in a Prescribed Set 528

13.3 Pexider Difference 530

13.3.1 Some Generalizations 532

13.3.2 Measurable Solutions of the Functional Equation (13.48a) 533

14 Characterization of Special Functions 537

14.1 Gamma Function 537

14.1.1 Further Properties of the Gamma Function 540

14.1.2 Definitions of the Function Γ(x) 541

14.2 Beta Function 549

14.2.1 Integral Representation of β 550

14.2.2 Other Special formulas 553

14.3 Riemann's Zeta Function 555

14.3.1 The Theta Function 556

14.4 Singular Functions 557

14.4.1 Cantor-Lebesgue Singular Function 558

14.4.2 Minkowski's Function 558

14.4.3 De Rham's Function 560

15 Miscellaneous Equations 563

15.1 A General Method: Method of Determinants 563

15.2 Means 570

15.2.1 Characterizations-Arithmetic and Exponential Means 575

15.2.2 Geometric Mean and the Root Mean Power 576

15.2.3 Stolarsky Mean 579

15.3 Some Comments about the Logarithmic Function 581

15.4 D'Alembert's Equation Revisited 588

15.4.1 Basic d'Alembert Functions 589

15.4.2 D'Alembert Groups: Examples 592

15.4.3 D'Alembert Groups: Generalities 594

15.4.4 Solvable Finite d'Alembert Groups 594

15.4.5 Nonsolvable Finite d'Alembert Groups 595

15.5 Polynomials Revisited 597

15.6 Inner Products Revisited 600

16 General Inequalities 607

16.1 Cauchy Functional Inequalities 607

16.2 Subadditive and Superadditive Functions 610

16.3 Logarithmic Inequality 612

16.4 Multiplicative Inequality 614

16.5 Convexity 614

16.6 Trigonometric Functional Inequality 617

16.7 Cosine and Sine Functional Inequalities 624

16.8 Functional Equation Concerning the Parallelogram Identity-Quadratic Inequality 626

16.9 Inequalities for the Gamma and Beta Functions via some Classical Results 627

16.9.1 Inequalities via Chebychev's Inequality 627

16.9.2 Inequalities via Hölder's Inequality 632

16.10 Simpson's Inequality and Applications 634

16.11 Applications for Special Means 636

16.12 Inequalities from Information Theory 638

16.12.1 Generalization 641

16.12.2 Application to Mixed Theory of Information 644

16.12.3 Continuous Solution of the Inequality (16.61) 648

16.13 Reproducing Scoring Systems 651

16.13.1 Solution of the Functional Inequality (16.66) 652

16.13.2 Symmetric Reproducing Scoring Systems 654

16.13.3 A Generalization of RSS 655

16.14 More Inequalities from Information Theory 656

16.15 Inequalities from Inner Product Spaces 658

16.16 Miscellaneous Inequalities 661

16.16.1 More on Convex Functions 661

16.16.2 Inequalities for Integrals 663

16.16.3 Cauchy-Schwarz-Hölder Inequalities 665

17 Applications 669

17.1 Binomial Expansion 669

17.2 Scalar or Dot Product 671

17.3 Economics 673

17.3.1 Duopoly Model 673

17.3.1.1 Duopoly Model I 673

17.3.1.2 Duopoly Model II 675

17.3.2 Cobb-Douglas (CD) Production Function and Quasilinearity 677

17.3.2.1 Quasilinearity 677

17.3.2.2 Determination of all Quasilinear, Linearly Homogeneous Functions 678

17.4 Business Mathematics-Interest 680

17.4.1 Interest Formula 680

17.4.2 Simple Interest 681

17.4.3 Interest Rates 681

17.5 Physics 682

17.5.1 Quantum Physics 682

17.5.2 Gaussian Function 684

17.5.3 Chebyshev Polynomials 688

17.5.3.1 Introduction 688

17.5.3.2 Reduction to a Difference Equation 689

17.5.3.3 The Universal Solution 691

17.5.3.4 Identification of the Universal Solution 692

17.5.3.5 General Remarks 693

17.6 Topology 694

17.6.1 Integral 697

17.6.1.1 Simpson's Rule 697

17.6.1.2 Solution of the Functional Equation (17.46) 699

17.6.1.3 Solution of the Functional Equation (17.47) 699

17.6.1.4 Solution of the Main Functional Equation (17.44) 703

17.6.2 Determinants 708

17.7 Digital Filtering 710

17.8 Geometry 711

17.9 Field Homomorphisms 713

17.10 Pythagorean Functional Equation 715

17.11 Statistics 717

17.11.1 Poisson Distribution 718

17.11.1.1 Characterization of (Bivariate) Poisson Distributions 718

17.11.2 Normal Distribution 719

17.11.2.1 The Equation f(x)g(y) = h(ax+by)k(cx+dy) 725

17.11.2.2 The Equation f(x)g(y) = $$ 728

17.11.2.3 Solution of the Functional Equation (17.72) 730

17.11.3 Gamma Distribution 733

17.12 Information Theory 734

17.12.1 Bose-Einstein Entropy 734

17.12.1.1 Solution of Equations (17.81) and (17.81a) 735

17.12.1.2 Solution of Equation (17.81b) 737

17.12.2 Sums of Powers 739

17.12.12.1 A General Result 739

17.13 Behavioural Sciences 745

17.13.1 A Behavioural Example 746

17.13.2 Psychophysics 748

17.13.2.1 The Conjoint Weber Law 749

17.13.3 Binocular Vision 750

17.13.3.1 The Luneburg Theory of Binocular Vision 751

17.13.3.2 A Conjoint Representation Generalizing the Luneburg Theory 752

17.13.4 Functional Equations Resulting from Psychophysical Invariances 753.

Notes:

Includes bibliographical references and indexes.

ISBN:

9780387894911

0387894918

OCLC:

310400909