Gautschi / Mastroianni / Rassias Approximation and Computation

In Honor of Gradimir V. Milovanovic
1. Auflage 2010
ISBN: 978-1-4419-6594-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 42, 482 Seiten

Reihe: Springer Optimization and Its Applications

Approximation theory and numerical analysis are central to the creation of accurate computer simulations and mathematical models. Research in these areas can influence the computational techniques used in a variety of mathematical and computational sciences. This collection of contributed chapters, dedicated to renowned mathematician Gradimir V. Milovanovic, represent the recent work of experts in the fields of approximation theory and numerical analysis. These invited contributions describe new trends in these important areas of research including theoretic developments, new computational algorithms, and multidisciplinary applications. Special features of this volume: - Presents results and approximation methods in various computational settings including: polynomial and orthogonal systems, analytic functions, and differential equations. - Provides a historical overview of approximation theory and many of its subdisciplines; - Contains new results from diverse areas of research spanning mathematics, engineering, and the computational sciences. 'Approximation and Computation' is intended for mathematicians and researchers focusing on approximation theory and numerical analysis, but can also be a valuable resource to students and researchers in the computational and applied sciences.

Autoren/Hrsg.

Gautschi, Walter

Mastroianni, Giuseppe

Rassias, Themistocles M.

Inhaltsverzeichnis

1;Preface;6
2;Contents
;10
3;Part I Introduction;20
3.1;The Scientific Work of Gradimir V. Milovanovic;21
3.1.1;1 Introduction;21
3.1.2;2 The Biography of G.V. Milovanovic;21
3.1.3;3 Fields of Scientific Work of GVM;23
3.1.4;4 GVM and Quadrature Processes;24
3.1.4.1;4.1 Construction of Gaussian Quadratures;24
3.1.4.2;4.2 Moment-Preserving Spline Approximation and Quadratures;26
3.1.4.3;4.3 Quadratures with Multiple Nodes;27
3.1.4.4;4.4 Orthogonality with Respect to a Moment Functional and Corresponding Quadratures;28
3.1.4.4.1;4.4.1 Orthogonality on the Semicircle and Quadratures;28
3.1.4.4.2;4.4.2 Orthogonal Polynomials for Oscillatory Weights;29
3.1.4.5;4.5 Nonstandard Quadratures of Gaussian Type;31
3.1.4.6;4.6 Gaussian Quadrature for Müntz Systems;33
3.1.5;5 GVM and Polynomials;34
3.1.5.1;5.1 Classical Orthogonal Polynomials;34
3.1.5.2;5.2 Extremal Problems of Markov–Bernstein Type for Polynomials;35
3.1.5.2.1;5.2.1 L2–Inequalities with Laguerre Measure for Nonnegative Polynomials;37
3.1.5.2.2;5.2.2 Extremal Problems for the Lorentz Class of Polynomials ;38
3.1.5.3;5.3 Orthogonal Polynomials on Radial Rays;39
3.1.6;6 GVM and Interpolation Processes;42
3.1.7;7 The Selected Bibliography of GVM;43
3.1.8;References;43
3.2;My Collaboration with Gradimir V. Milovanovic;51
3.2.1;References;60
3.3;On Some Major Trends in Mathematics;62
4;Part II Polynomials and Orthogonal Systems;67
4.1;An Application of Sobolev Orthogonal Polynomials to the Computation of a Special Hankel Determinant;68
4.1.1;1 Introduction;68
4.1.2;2 Preliminaries;69
4.1.3;3 The Properties of Number Sequences;70
4.1.4;4 The Hankel Determinants and Orthogonal Polynomials;71
4.1.5;5 Connections with Classical Orthogonal Polynomials;73
4.1.6;6 The Connection with Polynomials Orthogonal with Respect to a Discrete Sobolev Inner Product;74
4.1.7;References;75
4.2;Extremal Problems for Polynomials in the Complex Plane;76
4.2.1;1 Introduction;76
4.2.2;2 Gauss–Lucas Theorem and Extensions;78
4.2.3;3 Sendov's Conjecture;80
4.2.4;4 The Conjecture of Smale;88
4.2.5;5 Majorization of Polynomials;93
4.2.6;6 Bernstein's Inequality on Lemniscates;96
4.2.7;References;98
4.3;Energy of Graphs and Orthogonal Matrices;101
4.3.1;1 Introduction and Notation;101
4.3.2;2 Characterization by Projectors and Orthogonal Matrices;102
4.3.3;3 Some Applications;105
4.3.4;4 Conference Matrices and Asymptotic Behaviorof Maximal Energy;107
4.3.5;References;109
4.4;Interlacing Property of Zeros of Shifted Jacobi Polynomials;111
4.4.1;1 Introduction;111
4.4.2;2 Proof of Theorem 1;112
4.4.3;References;115
4.5;Trigonometric Orthogonal Systems;116
4.5.1;1 Introduction;116
4.5.2;2 Orthogonal Trigonometric Polynomials of Semi-Integer Degree;119
4.5.3;3 Recurrence Relations;121
4.5.4;4 Christoffel–Darboux Formula;124
4.5.5;5 Connection with Szegö Polynomials;125
4.5.6;6 Numerical Construction;126
4.5.7;References;128
4.6;Experimental Mathematics Involving Orthogonal Polynomials;130
4.6.1;1 Introduction;130
4.6.2;2 Inequalities for Zeros of Jacobi Polynomials;131
4.6.2.1;2.1 Inequalities for the Largest Zero;131
4.6.2.2;2.2 Inequalities for All Zeros;133
4.6.2.3;2.3 Modified Inequalities for All Zeros;134
4.6.3;3 Bernstein's Inequality for Jacobi Polynomials;136
4.6.3.1;3.1 Sharpness of Bernstein's Inequality;136
4.6.3.2;3.2 Bernstein's Inequality on Larger Domains;138
4.6.4;4 Quadrature Formulae;138
4.6.4.1;4.1 Positivity of Weighted Newton–Cotes Formulae;139
4.6.4.2;4.2 Positivity of Generalized Gauss–Radau Formulae;140
4.6.4.3;4.3 Positivity of Generalized Gauss–Lobatto Formulae;142
4.6.4.4;4.4 Positivity of Most General Gauss–Radau/Lobatto Formulae;142
4.6.5;5 Gauss Quadrature with Exotic Weight Functions;143
4.6.5.1;5.1 Weight Function Decaying Super-Exponentially at Infinity;143
4.6.5.2;5.2 Weight Functions Densely Oscillating at Zero;144
4.6.6;References;146
4.7;Compatibility of Continued Fraction Convergents with Padé Approximants;148
4.7.1;1 Introduction;148
4.7.2;2 Classical Continued Fractions and Their Relations with Padé Approximants;149
4.7.3;3 Method of Analysis and Simplified Notation;151
4.7.4;4 Main Results;152
4.7.5;Reference;157
4.8;Orthogonal Decomposition of Fractal Sets;158
4.8.1;1 Introduction;158
4.8.2;2 Hyperplane of Areal Coordinates;159
4.8.3;3 Changing AIFS to IFS and Back;166
4.8.4;References;169
4.9;Positive Trigonometric Sums and Starlike Functions;170
4.9.1;1 Introduction;170
4.9.2;2 Positive Trigonometric Sums;171
4.9.2.1;2.1 Vietoris' Inequalities;172
4.9.3;3 Extensions of Vietoris' Cosine Inequality;174
4.9.3.1;3.1 Remarks;175
4.9.3.2;3.2 Further Generalizations and Related Results;176
4.9.4;4 Starlike Functions;184
4.9.4.1;4.1 Positive Sums of Gegenbauer Polynomials;186
4.9.4.2;4.2 Subordination and Convolution of Analytic Functions;187
4.9.5;5 Generalizations and Extensions;190
4.9.6;References;194
5;Part III Quadrature Formulae;196
5.1;Quadrature Rules for Unbounded Intervals and Their Application to Integral Equations;197
5.1.1;1 Introduction;197
5.1.2;2 Quadrature Formulae on Half-Infinite Interval (0,);198
5.1.3;3 Quadrature Formulae on Infinite Interval (-,)
;207
5.1.4;4 An Application to Some Integral Equations;217
5.1.5;5 Conclusions;219
5.1.6;References;220
5.2;Gauss-Type Quadrature Formulae for Parabolic Splines with Equidistant Knots;221
5.2.1;1 Introduction;221
5.2.2;2 Spline Functions and Peano Kernels of Quadratures;224
5.2.3;3 Gaussian Quadrature Formulae for Parabolic Splines;225
5.2.3.1;3.1 The Construction;225
5.2.3.2;3.2 A Formula for c3,(QnG);229
5.2.3.3;3.3 Estimates for c3,(QnG);232
5.2.4;4 Lobatto Quadrature Formulae for Parabolic Splines;234
5.2.4.1;4.1 The Construction;234
5.2.4.2;4.2 Estimates for c3,(QnL);237
5.2.5;5 Concluding Remarks;241
5.2.6;References;243
5.3;Approximation of the Hilbert Transform on the Real Line Using Freud Weights;244
5.3.1;1 Introduction;244
5.3.2;2 Preliminary Results and Notations;245
5.3.2.1;2.1 Function Spaces;245
5.3.2.2;2.2 Orthonormal Polynomials and Gaussian Rule;247
5.3.3;3 Main Results;248
5.3.4;4 Numerical Examples;251
5.3.5;5 Proofs;253
5.3.6;References;262
5.4;The Remainder Term of Gauss–Turán Quadratures for Analytic Functions;264
5.4.1;1 Introduction;264
5.4.2;2 Error Bounds of Type (5);267
5.4.3;3 Error Bounds of the Type (6);272
5.4.4;4 Practical Error Estimates;273
5.4.5;References;276
5.5;Towards a General Error Theory of the Trapezoidal Rule;278
5.5.1;1 Introduction;278
5.5.2;2 The Classical Trapezoidal Rule;279
5.5.3;3 The Periodic Case;280
5.5.4;4 Integrals Over the Real Line;281
5.5.5;5 Transforming the Integration Variable;283
5.5.6;6 Error Theory of Integrals Over R;284
5.5.7;7 Asymptotics;285
5.5.7.1;7.1 An Introductory Example;286
5.5.7.2;7.2 A Difficult Integral;289
5.5.8;8 Conclusions;292
5.5.9;References;292
6;Part IV Differential Equations;294
6.1;Finite Difference Method for a Parabolic Problem with Concentrated Capacity and Time-Dependent Operator;295
6.1.1;1 Introduction;295
6.1.2;2 Preliminary Results;296
6.1.3;3 Heat Equation with Concentrated Capacity;299
6.1.4;4 The Difference Problem;300
6.1.5;5 Convergence of the Difference Scheme;301
6.1.6;References;306
6.2;Adaptive Finite Element Approximation of the Francfort–Marigo Model of Brittle Fracture;307
6.2.1;1 Introduction;307
6.2.1.1;1.1 The Francfort–Marigo Model of Brittle Fracture;308
6.2.1.2;1.2 The Ambrosio-Tortorelli Approximation;310
6.2.1.3;1.3 Critical Points;311
6.2.2;2 Adaptive Finite Element Discretization;312
6.2.2.1;2.1 The Alternating Minimization Algorithm;312
6.2.2.2;2.2 Finite Element Discretization;314
6.2.2.3;2.3 Adaptive Alternating Minimization;315
6.2.3;3 A Computational Example;317
6.2.4;References;320
6.3;A Nyström Method for Solving a Boundary Value Problem on [0,);321
6.3.1;1 Introduction;321
6.3.2;2 Function Spaces and Preliminary Results;322
6.3.3;3 Numerical Method;324
6.3.4;4 Numerical Examples;326
6.3.5;5 Proofs;329
6.3.6;References;335
6.4;Finite Difference Approximation of a Hyperbolic TransmissionProblem;336
6.4.1;1 Introduction;336
6.4.2;2 Formulation of the Initial Boundary Value Problem;337
6.4.3;3 Existence and Uniqueness of Weak Solution;338
6.4.4;4 Finite Difference Approximation;340
6.4.4.1;4.1 Meshes and Finite Difference Operators;340
6.4.4.2;4.2 Finite Difference Scheme;341
6.4.4.3;4.3 Convergence of the Finite Difference Scheme;342
6.4.5;References;346
6.5;Homeomorphisms and Fredholm Theory for Perturbations of Nonlinear Fredholm Maps of Index Zero and of A-Proper Maps with Applications;347
6.5.1;1 Part I. Existence Theory;347
6.5.1.1;1.1 Perturbations of Homeomorphisms and Nonlinear Fredholm Alternatives;347
6.5.1.2;1.2 Finite Solvability of Equations with Perturbations of Odd Fredholm Maps of Index Zero;352
6.5.1.3;1.3 Applications to (Quasi) Linear Elliptic Nonlinear Boundary Value Problems;354
6.5.2;2 Part II. Constructive Theory;358
6.5.2.1;2.1 Constructive Homeomorphism Results and Error Estimates;358
6.5.2.2;2.2 Constructive Homeomorphisms andTheir Perturbations;363
6.5.2.3;2.3 Nonlinear Alternatives for Perturbations of A-Proper Homeomorphisms;365
6.5.3;References;369
6.6;Singular Support and FLq Continuity of Pseudodifferential Operators;372
6.6.1;1 Introduction;372
6.6.2;2 Notions and Notation;373
6.6.3;3 Wave-Front Sets in FLq;375
6.6.4;4 Convolution in FLq;377
6.6.5;5 Multiplication in F Lqs (Rd);380
6.6.6;6 Continuity of Pseudodifferential Operators on FLq;382
6.6.7;7 Pseudodifferential Operators, an Extension;387
6.6.8;References;389
6.7;On a Class of Matrix Differential Equations with PolynomialCoefficients;391
6.7.1;1 Introduction;391
6.7.2;2 Main Result;394
6.7.3;References;396
7;Part V Applications;397
7.1;Optimized Algorithm for Petviashvili's Method for Finding Solitons in Photonic Lattices;398
7.1.1;1 Introduction;398
7.1.2;2 Physical Model;399
7.1.3;3 Modified Petviashvili's Method for Finding Solitonic Solutions in Photonic Lattices;399
7.1.4;4 Software Simulator;400
7.1.5;5 Conclusion;404
7.1.6;References;405
7.2;Explicit Method for the Numerical Solution of the Fokker-Planck Equation of Filtered Phase Noise;406
7.2.1;1 Introduction;406
7.2.2;2 Filtering Model;407
7.2.3;3 Application of FP Equation;408
7.2.4;4 Numerical Procedure;409
7.2.5;5 Numerical Results;411
7.2.6;6 Conclusion;412
7.2.7;References;412
7.3;Numerical Method for Computer Study of Liquid Phase Sintering: Densification Due to Gravity-Induced Skeletal Settling;413
7.3.1;1 Introduction;413
7.3.2;2 Simulation Model;414
7.3.2.1;2.1 Initial Configuration;414
7.3.2.2;2.2 Settling Procedure;415
7.3.2.3;2.3 Solid Skeleton Topology;418
7.3.2.4;2.4 Solution-Reprecipitation;421
7.3.3;3 Result and Discussion;423
7.3.4;4 Conclusion;426
7.3.5;References;427
7.4;Computer Algebra and Line Search;429
7.4.1;1 Introduction;429
7.4.2;2 Preliminaries;431
7.4.3;3 Implementation of Exact Line Search;433
7.4.4;4 Numerical Results and Comparisons;436
7.4.5;5 Conclusion;438
7.4.6;Appendix;439
7.4.7;References;442
7.5;Roots of AG-bands;443
7.5.1;1 Introduction;443
7.5.2;2 Subclasses of Roots of a Band;444
7.5.3;References;449
7.6;Context Hidden Markov Model for Named Entity Recognition;450
7.6.1;1 Introduction;450
7.6.2;2 Word-Feature Pairs;452
7.6.3;3 Context Hidden Markov Model for Named Entity Recognition;452
7.6.3.1;3.1 Hidden Markov Model in NERC;452
7.6.3.2;3.2 Context Hidden Markov Model in NERC;455
7.6.4;4 Training;456
7.6.5;5 The Sparseness of Data and the Expectation Maximization;457
7.6.6;6 Expectation Maximization;458
7.6.7;7 Grammar Component of the NERC System;460
7.6.8;8 Experimental Results;461
7.6.9;References;463
7.7;On the Interpolating Quadratic Spline;464
7.7.1;1 Introduction;464
7.7.2;2 Basic Results;466
7.7.3;3 A Note;470
7.7.4;References;470
7.8;Visualization of Infinitesimal Bending of Curves;471
7.8.1;1 Preliminaries;471
7.8.2;2 Infinitesimal Bending in the Plane;473
7.8.3;3 Variation of Torsion and Curvature;475
7.8.4;4 InfBend;478
7.8.5;References;481