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Orthogonal Polynomials: Computation and Approximation
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关于正交多项式在计算中的应用于方法论CONTENTS Preface viii 1 Basic Theory 1 1.1 Orthogonal polynomials 1 1.1.1 Definition and existence 1 1.1.2 Examples 4 1.2 Properties of orthogonal polynomials 6 1.2.1 Symmetry 6 1.2.2 Zeros 7 1.2.3 Discrete orthogonality 8 1.2.4 Extremal properties 8 1.3 Three-term recurrence relation 10 1.3.1 Monic orthogonal polynomials 10 1.3.2 Orthonormal polynomials 12 1.3.3 Christoffel–Darboux formulae 14 1.3.4 Continued fractions 15 1.3.5 The recurrence relation outside the support interval 17 1.4 Quadrature rules 20 1.4.1 Interpolatory quadrature rules and beyond 21 1.4.2 Gauss-type quadrature rules 22 1.5 Classical orthogonal polynomials 26 1.5.1 Classical orthogonal polynomials of a continuous variable 27 1.5.2 Classical orthogonal polynomials of a discrete variable 32 1.6 Kernel polynomials 35 1.6.1 Existence and elementary properties 36 1.6.2 Recurrence relation 38 1.7 Sobolev orthogonal polynomials 40 1.7.1 Definition and properties 41 1.7.2 Recurrence relation and zeros 41 1.8 Orthogonal polynomials on the semicircle 43 1.8.1 Definition, existence, and representation 43 1.8.2 Recurrence relation 45 1.8.3 Zeros 47 1.9 Notes to Chapter 1 49 v vi CONTENTS 2 Computational Methods 52 2.1 Moment-based methods 52 2.1.1 Classical approach via moment determinants 52 2.1.2 Condition of nonlinear maps 55 2.1.3 The moment maps Gn and Kn 57 2.1.4 Condition of Gn : μ 7! 59 2.1.5 Condition of Gn : m 7! 64 2.1.6 Condition of Kn : m 7! 70 2.1.7 Modified Chebyshev algorithm 76 2.1.8 Finite expansions in orthogonal polynomials 78 2.1.9 Examples 82 2.2 Discretization methods 90 2.2.1 Convergence of discrete orthogonal polynomials to continuous ones 90 2.2.2 A general-purpose discretization procedure 93 2.2.3 Computing the recursion coefficients of a discrete measure 95 2.2.4 A multiple-component discretization method 99 2.2.5 Examples 101 2.2.6 Discretized modified Chebyshev algorithm 111 2.3 Computing Cauchy integrals of orthogonal polynomials 112 2.3.1 Characterization in terms of minimal solutions 112 2.3.2 A continued fraction algorithm 113 2.3.3 Examples 116 2.4 Modification algorithms 121 2.4.1 Christoffel and generalized Christoffel theorems 122 2.4.2 Linear factors 124 2.4.3 Quadratic factors 125 2.4.4 Linear divisors 128 2.4.5 Quadratic divisors 130 2.4.6 Examples 133 2.5 Computing Sobolev orthogonal polynomials 138 2.5.1 Algorithm based on moment information 139 2.5.2 Stieltjes-type algorithm 141 2.5.3 Zeros 143 2.5.4 Finite expansions in Sobolev orthogonal polynomials 146 2.6 Notes to Chapter 2 148 3 Applications 152 3.1 Quadrature 152 3.1.1 Computation of Gauss-type quadrature formulae 152 CONTENTS vii 3.1.2 Gauss–Kronrod quadrature formulae and their computation 165 3.1.3 Gauss–Tur´an quadrature formulae and their computation 172 3.1.4 Quadrature formulae based on rational functions 180 3.1.5 Cauchy principal value integrals 202 3.1.6 Polynomials orthogonal on several intervals 207 3.1.7 Quadrature estimation of matrix functionals 211 3.2 Least squares approximation 216 3.2.1 Classical least squares approximation 217 3.2.2 Constrained least squares approximation 221 3.2.3 Least squares approximation in Sobolev spaces 225 3.3 Moment-preserving spline approximation 227 3.3.1 Approximation on the positive real line 228 3.3.2 Approximation on a compact interval 237 3.4 Slowly convergent series 239 3.4.1 Series generated by a Laplace transform 240 3.4.2 “Alternating” series generated by a Laplace transform 245 3.4.3 Series generated by the derivative of a Laplace transform 246 3.4.4 “Alternating” series generated by the derivative of a Laplace transform 248 3.4.5 Slowly convergent series occurring in plate contact problems 249 3.5 Notes to Chapter 3 253 Bibliography 261 Index 283 [ Last edited by feixiaolin on 2014-1-5 at 07:27 ] |
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