Interpolation Processes (eBook)

Basic Theory and Applications

Giuseppe Mastroianni, Gradimir Milovanovic (Autoren)

eBook Download: PDF

2008 | 1. Auflage
446 Seiten
Springer-Verlag
978-3-540-68349-0 (ISBN)

Infos

Interpolation of functions is one of the basic part of Approximation Theory. There are many books on approximation theory, including interpolation methods that - peared in the last fty years, but a few of them are devoted only to interpolation processes. An example is the book of J. Szabados and P. Vértesi: Interpolation of Functions, published in 1990 by World Scienti c. Also, two books deal with a special interpolation problem, the so-called Birkhoff interpolation, written by G.G. Lorentz, K. Jetter, S.D. Riemenschneider (1983) and Y.G. Shi (2003). The classical books on interpolation address numerous negative results, i.e., - sultsondivergentinterpolationprocesses,usuallyconstructedoversomeequidistant system of nodes. The present book deals mainly with new results on convergent - terpolation processes in uniform norm, for algebraic and trigonometric polynomials, not yet published in other textbooks and monographs on approximation theory and numerical mathematics. Basic tools in thiseld (orthogonal polynomials, moduli of smoothness,K-functionals, etc.), as well as some selected applications in numerical integration, integral equations, moment-preserving approximation and summation of slowly convergent series are also given. The rstchapterprovidesanaccountofbasicfactsonapproximationbyalgebraic and trigonometric polynomials introducing the most important concepts on appro- mation of functions. Especially, in Sect. 1.4 we give basic results on interpolation by algebraic polynomials, including representations and computation of interpolation polynomials, Lagrange operators, interpolation errors and uniform convergence in some important classes of functions, as well as an account on the Lebesgue function and some estimates for the Lebesgue constant.

Autor

Gradimir V. Milovanovic is Professor of the University of Niš and Corresponding member of the Serbian Academy of Sciences and Arts.

Inhalt

Preface 6
Contents 9
Constructive Elements and Approaches in Approximation Theory 13
Introduction to Approximation Theory 13
Basic Notions 13
Algebraic and Trigonometric Polynomials 16
Best Approximation by Polynomials 19
Chebyshev Polynomials 21
Basic Properties 21
Differential Equation 22
Zeros and Extremal Points 23
Chebyshev Polynomials in the Complex Plane 24
Some Other Relations 25
Orthogonality 26
Chebyshev Extremal Problems 26
The Extremal Problem in the Uniform Norm 26
The Extremal Problem in L1-norm 28
Chebyshev Alternation Theorem 29
Some Classical Special Cases 31
Numerical Methods 32
Basic Facts on Trigonometric Approximation 36
Trigonometric Kernels 36
Fourier Series and Sums 42
Moduli of Smoothness, Best Approximation and Besov Spaces 44
Chebyshev Systems and Interpolation 50
Chebyshev Systems and Spaces 50
Algebraic Lagrange Interpolation 51
Trigonometric Interpolation 52
Riesz Interpolation Formula 56
A General Interpolation Problem 58
Interpolation by Algebraic Polynomials 60
Representations and Computation of Interpolation Polynomials 60
Interpolation Array and Lagrange Operators 63
Interpolation Error for Some Classes of Functions 66
The Error in the Class of Continuous-Differentiable Functions 66
The Error in the Class of Analytic Functions 67
Uniform Convergence in the Class of Analytic Functions 68
Bernstein's Example of Pointwise Divergence 73
Lebesgue Function and Some Estimates for the Lebesgue Constant 75
Equidistant Nodes 76
Chebyshev Nodes 77
Algorithm for Finding Optimal Nodes 80
Orthogonal Polynomials and Weighted Polynomial Approximation 86
Orthogonal Systems and Polynomials 86
Inner Product Space and Orthogonal Systems 86
Fourier Expansion and Best Approximation 88
Examples of Orthogonal Systems 90
Trigonometric System 90
Chebyshev Polynomials 90
Orthogonal Polynomials on the Unit Circle 91
Orthogonal Polynomials on the Unit Disk 91
Orthogonal Polynomials on the Ellipse 91
Malmquist-Takenaka System of Rational Functions 92
Polynomials Orthogonal on the Radial Rays 92
Müntz Orthogonal Polynomials 93
Müntz Orthogonal Polynomials of the Second Kind 95
Generalized Exponential Polynomials 96
Discrete Chebyshev Polynomials 96
Formal Orthogonal Polynomials with Respect to a Moment Functional 97
Basic Facts on Orthogonal Polynomials and Extremal Problems 100
Zeros of Orthogonal Polynomials 104
Orthogonal Polynomials on the Real Line 106
Basic Properties 106
Three-Term Recurrence Relations 107
Christoffel's Formulae 109
Zeros 110
Some Special Weights 112
Asymptotic Properties of Orthogonal Polynomials 114
Bernstein-Szego Identities 119
The Fokas-Its-Kitaev (Riemann-Hilbert) Identity 120
Rakhmanov's Identity 122
Associated Polynomials and Christoffel Numbers 122
Associated Polynomials 122
Stieltjes Transform of the Measure and Christoffel Numbers 125
Markov's Moment Problem 127
Functions of the Second Kind and Stieltjes Polynomials 128
Classical Orthogonal Polynomials 132
Definition of the Classical Orthogonal Polynomials 132
General Properties of the Classical Orthogonal Polynomials 135
Generating Function 139
Jacobi Polynomials 142
Special Cases 144
Zeros 146
Inequalities and Asymptotics 147
Christoffel Function and Christoffel Numbers 150
Generalized Laguerre Polynomials 151
Zeros 152
Inequalities 153
Christoffel Function and Christoffel Numbers 155
Hermite Polynomials 156
Nonclassical Orthogonal Polynomials 157
Semi-classical Orthogonal Polynomials 157
Generalized Gegenbauer Polynomials 158
Generalized Jacobi Polynomials 159
Sonin-Markov Orthogonal Polynomials 163
Freud Orthogonal Polynomials 165
Mhaskar-Rakhmanov-Saff Number 165
Basic Properties of Freud Polynomials 166
Strong Asymptotics 168
Orthogonal Polynomials with Respect to Abel, Lindelöf, and Logistic Weights 170
Strong Non-classical Orthogonal Polynomials 170
Numerical Construction of Orthogonal Polynomials 171
Modified Chebyshev Algorithm 171
Discretized Stieltjes-Gautschi Procedure 173
Weighted Polynomial Approximation 177
Weighted Functional Spaces, Moduli of Smoothness and K-functionals 177
Weighted Best Polynomial Approximation on [-1,1] 181
Weighted Approximation on the Semi-axis 185
Weighted K-functionals and Moduli of Smoothness 186
Weighted Best Polynomial Approximation 187
Weighted Besov Type Spaces 188
Weighted Approximation on the Real Line 189
Weighted Polynomial Approximation of Functions Having Isolated Interior Singularities 193
Trigonometric Approximation 204
Approximating Properties of Operators 204
Approximation by Fourier Sums 204
Approximation by Fejér and de la Vallée Poussin Means 206
Discrete Operators 208
A Quadrature Formula 208
Discrete Versions of Fourier and de la Vallée Poussin Sums 213
Marcinkiewicz Inequalities 216
Uniform Approximation 221
Lagrange Interpolation Error in Lp 223
Some Estimates of the Interpolation Errors in L1-Sobolev Spaces 232
The Weighted Case 235
Algebraic Interpolation in Uniform Norm 245
Introduction and Preliminaries 245
Interpolation at Zeros of Orthogonal Polynomials 245
Some Auxiliary Results 249
Optimal Systems of Nodes 258
Optimal Systems of Knots on [-1,1] 258
Interpolation at Jacobi Abscissas 258
Interpolation at the ``Practical Abscissas'' 259
Additional Nodes Method with Jacobi Zeros 262
Other ``Optimal'' Interpolation Processes 274
Interpolation with Associated Polynomials 274
Interpolation at Stieltjes Zeros 276
Extended Interpolation 276
Some Simultaneous Interpolation Processes 278
Weighted Interpolation 281
Weighted Interpolation at Jacobi Zeros 281
Lagrange Interpolation in Sobolev Spaces 286
Interpolation at Laguerre Zeros 288
Interpolation at Hermite Zeros 297
Interpolation of Functions with Internal Isolated Singularities 302
Interpolation Processes on Bounded Intervals 305
Interpolation Processes on Unbounded Intervals 316
Numerical Examples 319
Applications 329
Quadrature Formulae 329
Introduction 329
Some Remarks on Newton-Cotes Rules with Jacobi Weights 332
Gauss-Christoffel Quadrature Rules 334
Gauss-Christoffel Quadratures for the Classical Weights 334
Computation of Gauss-Christoffel Quadratures 335
Gauss-Radau and Gauss-Lobatto Quadrature Rules 338
Gauss-Radau Quadrature Formula 339
Gauss-Lobatto Quadrature Formula 340
Error Estimates of Gaussian Rules for Some Classes of Functions 342
Error Estimates for Analytic Functions 344
Error Estimates for Some Classes of Continuous Functions 347
Error Estimates for Gauss-Laguerre Formula 351
Error Estimates for Freud-Gaussian Rules 353
Product Integration Rules 355
Integration of Periodic Functions on the Real Line with Rational Weight 360
Integral Equations 372
Some Basic Facts 372
Fredholm Integral Equations of the Second Kind 379
Locally Smooth Kernels 380
Numerical Examples 386
Weakly Singular Kernels 389
Nyström Method 392
Moment-Preserving Approximation 395
The Standard L2-Approximation 395
Generalization 397
The Constrained L2-Polynomial Approximation 398
Moment-Preserving Spline Approximation 399
Approximation on [0,+) 399
Approximation on a Compact Interval 405
Summation of Slowly Convergent Series 407
Laplace Transform Method 408
Contour Integration Over a Rectangle 411
Remarks on Some Slowly Convergent Power Series 421
References 424
Index 445

Produkt-Details

• Erscheint lt. Verlag: 24.8.2008
• Reihe/Serie: Springer Monographs in Mathematics
• Reihe/Serie: Springer Monographs in Mathematics
• Zusatzinfo: XIV, 446 p. 42 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Schlagworte: Gaussian quadratures • integral equation • Integral equations • Interpolation • MSC(2000): 33-xx, 41-xx, 42Axx, 45A05, 45B05, 45H05, 65B10, 65Dxx • orthogonal polynomials • polynomial approximation • Real analysis • Sobolev Space
• ISBN-10: 3-540-68349-6 / 3540683496
• ISBN-13: 978-3-540-68349-0 / 9783540683490