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Approximation Theory and Methods - M. J. D. Powell

Approximation Theory and Methods

Paperback

Published: 20th July 1981
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Most functions that occur in mathematics cannot be used directly in computer calculations. Instead they are approximated by manageable functions such as polynomials and piecewise polynomials. The general theory of the subject and its application to polynomial approximation are classical, but piecewise polynomials have become far more useful during the last twenty years. Thus many important theoretical properties have been found recently and many new techniques for the automatic calculation of approximations to prescribed accuracy have been developed. This book gives a thorough and coherent introduction to the theory that is the basis of current approximation methods. Professor Powell describes and analyses the main techniques of calculation supplying sufficient motivation throughout the book to make it accessible to scientists and engineers who require approximation methods for practical needs. Because the book is based on a course of lectures to third-year undergraduates in mathematics at Cambridge University, sufficient attention is given to theory to make it highly suitable as a mathematical textbook at undergraduate or postgraduate level.

"Of its kind, this book is excellent, perhaps the best." Journal of Classification

Prefacep. ix
The approximation problem and existence of best approximationsp. 1
Examples of approximation problemsp. 1
Approximation in a metric spacep. 3
Approximation in a normed linear spacep. 5
The L[subscript p]-normsp. 6
A geometric view of best approximationsp. 9
The uniqueness of best approximationsp. 13
Convexity conditionsp. 13
Conditions for the uniqueness of the best approximationp. 14
The continuity of best approximation operatorsp. 16
The 1-, 2- and [infinity]-normsp. 17
Approximation operators and some approximating functionsp. 22
Approximation operatorsp. 22
Lebesgue constantsp. 24
Polynomial approximations to differentiable functionsp. 25
Piecewise polynomial approximationsp. 28
Polynomial interpolationp. 33
The Lagrange interpolation formulap. 33
The error in polynomial interpolationp. 35
The Chebyshev interpolation pointsp. 37
The norm of the Lagrange interpolation operatorp. 41
Divided differencesp. 46
Basic properties of divided differencesp. 46
Newton's interpolation methodp. 48
The recurrence relation for divided differencesp. 49
Discussion of formulae for polynomial interpolationp. 51
Hermite interpolationp. 53
The uniform convergence of polynomial approximationsp. 61
The Weierstrass theoremp. 61
Monotone operatorsp. 62
The Bernstein operatorp. 65
The derivatives of the Bernstein approximationsp. 67
The theory of minimax approximationp. 72
Introduction to minimax approximationp. 72
The reduction of the error of a trial approximationp. 74
The characterization theorem and the Haar conditionp. 76
Uniqueness and bounds on the minimax errorp. 79
The exchange algorithmp. 85
Summary of the exchange algorithmp. 85
Adjustment of the referencep. 87
An example of the iterations of the exchange algorithmp. 88
Applications of Chebyshev polynomials to minimax approximationp. 90
Minimax approximation on a discrete point setp. 92
The convergence of the exchange algorithmp. 97
The increase in the levelled reference errorp. 97
Proof of convergencep. 99
Properties of the point that is brought into referencep. 102
Second-order convergencep. 105
Rational approximation by the exchange algorithmp. 111
Best minimax rational approximationp. 111
The best approximation on a referencep. 113
Some convergence properties of the exchange algorithmp. 116
Methods based on linear programmingp. 118
Least squares approximationp. 123
The general form of a linear least squares calculationp. 123
The least squares characterization theoremp. 125
Methods of calculationp. 126
The recurrence relation for orthogonal polynomialsp. 131
Properties of orthogonal polynomialsp. 136
Elementary propertiesp. 136
Gaussian quadraturep. 138
The characterization of orthogonal polynomialsp. 141
The operator R[subscript n]p. 143
Approximation to periodic functionsp. 150
Trigonometric polynomialsp. 150
The Fourier series operator S[subscript n]p. 152
The discrete Fourier series operatorp. 156
Fast Fourier transformsp. 158
The theory of best L[subscript 1] approximationp. 164
Introduction to best L[subscript 1] approximationp. 164
The characterization theoremp. 165
Consequences of the Haar conditionp. 169
The L[subscript 1] interpolation points for algebraic polynomialsp. 172
An example of L[subscript 1] approximation and the discrete casep. 177
A useful example of L[subscript 1] approximationp. 177
Jackson's first theoremp. 179
Discrete L[subscript 1] approximationp. 181
Linear programming methodsp. 183
The order of convergence of polynomial approximationsp. 189
Approximations to non-differentiable functionsp. 189
The Dini-Lipschitz theoremp. 192
Some bounds that depend on higher derivativesp. 194
Extensions to algebraic polynomialsp. 195
The uniform boundedness theoremp. 200
Preliminary resultsp. 200
Tests for uniform convergencep. 202
Application to trigonometric polynomialsp. 204
Application to algebraic polynomialsp. 208
Interpolation by piecewise polynomialsp. 212
Local interpolation methodsp. 212
Cubic spline interpolationp. 215
End conditions for cubic spline interpolationp. 219
Interpolating splines of other degreesp. 221
B-splinesp. 227
The parameters of a spline functionp. 227
The form of B-splinesp. 229
B-splines as basis functionsp. 231
A recurrence relation for B-splinesp. 234
The Schoenberg-Whitney theoremp. 236
Convergence properties of spline approximationsp. 241
Uniform convergencep. 241
The order of convergence when f is differentiablep. 243
Local spline interpolationp. 246
Cubic splines with constant knot spacingp. 248
Knot positions and the calculation of spline approximationsp. 254
The distribution of knots at a singularityp. 254
Interpolation for general knotsp. 257
The approximation of functions to prescribed accuracyp. 261
The Peano kernel theoremp. 268
The error of a formula for the solution of differential equationsp. 268
The Peano kernel theoremp. 270
Application to divided differences and to polynomial interpolationp. 274
Application to cubic spline interpolationp. 277
Natural and perfect splinesp. 283
A variational problemp. 283
Properties of natural splinesp. 285
Perfect splinesp. 290
Optimal interpolationp. 298
The optimal interpolation problemp. 298
L[subscript 1] approximation by B-splinesp. 301
Properties of optimal interpolationp. 307
The Haar conditionp. 313
Related work and referencesp. 317
Indexp. 333
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521295147
ISBN-10: 0521295149
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 352
Published: 20th July 1981
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24  x 1.91
Weight (kg): 0.48