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
Preface | p. ix |
The approximation problem and existence of best approximations | p. 1 |
Examples of approximation problems | p. 1 |
Approximation in a metric space | p. 3 |
Approximation in a normed linear space | p. 5 |
The L[subscript p]-norms | p. 6 |
A geometric view of best approximations | p. 9 |
The uniqueness of best approximations | p. 13 |
Convexity conditions | p. 13 |
Conditions for the uniqueness of the best approximation | p. 14 |
The continuity of best approximation operators | p. 16 |
The 1-, 2- and [infinity]-norms | p. 17 |
Approximation operators and some approximating functions | p. 22 |
Approximation operators | p. 22 |
Lebesgue constants | p. 24 |
Polynomial approximations to differentiable functions | p. 25 |
Piecewise polynomial approximations | p. 28 |
Polynomial interpolation | p. 33 |
The Lagrange interpolation formula | p. 33 |
The error in polynomial interpolation | p. 35 |
The Chebyshev interpolation points | p. 37 |
The norm of the Lagrange interpolation operator | p. 41 |
Divided differences | p. 46 |
Basic properties of divided differences | p. 46 |
Newton's interpolation method | p. 48 |
The recurrence relation for divided differences | p. 49 |
Discussion of formulae for polynomial interpolation | p. 51 |
Hermite interpolation | p. 53 |
The uniform convergence of polynomial approximations | p. 61 |
The Weierstrass theorem | p. 61 |
Monotone operators | p. 62 |
The Bernstein operator | p. 65 |
The derivatives of the Bernstein approximations | p. 67 |
The theory of minimax approximation | p. 72 |
Introduction to minimax approximation | p. 72 |
The reduction of the error of a trial approximation | p. 74 |
The characterization theorem and the Haar condition | p. 76 |
Uniqueness and bounds on the minimax error | p. 79 |
The exchange algorithm | p. 85 |
Summary of the exchange algorithm | p. 85 |
Adjustment of the reference | p. 87 |
An example of the iterations of the exchange algorithm | p. 88 |
Applications of Chebyshev polynomials to minimax approximation | p. 90 |
Minimax approximation on a discrete point set | p. 92 |
The convergence of the exchange algorithm | p. 97 |
The increase in the levelled reference error | p. 97 |
Proof of convergence | p. 99 |
Properties of the point that is brought into reference | p. 102 |
Second-order convergence | p. 105 |
Rational approximation by the exchange algorithm | p. 111 |
Best minimax rational approximation | p. 111 |
The best approximation on a reference | p. 113 |
Some convergence properties of the exchange algorithm | p. 116 |
Methods based on linear programming | p. 118 |
Least squares approximation | p. 123 |
The general form of a linear least squares calculation | p. 123 |
The least squares characterization theorem | p. 125 |
Methods of calculation | p. 126 |
The recurrence relation for orthogonal polynomials | p. 131 |
Properties of orthogonal polynomials | p. 136 |
Elementary properties | p. 136 |
Gaussian quadrature | p. 138 |
The characterization of orthogonal polynomials | p. 141 |
The operator R[subscript n] | p. 143 |
Approximation to periodic functions | p. 150 |
Trigonometric polynomials | p. 150 |
The Fourier series operator S[subscript n] | p. 152 |
The discrete Fourier series operator | p. 156 |
Fast Fourier transforms | p. 158 |
The theory of best L[subscript 1] approximation | p. 164 |
Introduction to best L[subscript 1] approximation | p. 164 |
The characterization theorem | p. 165 |
Consequences of the Haar condition | p. 169 |
The L[subscript 1] interpolation points for algebraic polynomials | p. 172 |
An example of L[subscript 1] approximation and the discrete case | p. 177 |
A useful example of L[subscript 1] approximation | p. 177 |
Jackson's first theorem | p. 179 |
Discrete L[subscript 1] approximation | p. 181 |
Linear programming methods | p. 183 |
The order of convergence of polynomial approximations | p. 189 |
Approximations to non-differentiable functions | p. 189 |
The Dini-Lipschitz theorem | p. 192 |
Some bounds that depend on higher derivatives | p. 194 |
Extensions to algebraic polynomials | p. 195 |
The uniform boundedness theorem | p. 200 |
Preliminary results | p. 200 |
Tests for uniform convergence | p. 202 |
Application to trigonometric polynomials | p. 204 |
Application to algebraic polynomials | p. 208 |
Interpolation by piecewise polynomials | p. 212 |
Local interpolation methods | p. 212 |
Cubic spline interpolation | p. 215 |
End conditions for cubic spline interpolation | p. 219 |
Interpolating splines of other degrees | p. 221 |
B-splines | p. 227 |
The parameters of a spline function | p. 227 |
The form of B-splines | p. 229 |
B-splines as basis functions | p. 231 |
A recurrence relation for B-splines | p. 234 |
The Schoenberg-Whitney theorem | p. 236 |
Convergence properties of spline approximations | p. 241 |
Uniform convergence | p. 241 |
The order of convergence when f is differentiable | p. 243 |
Local spline interpolation | p. 246 |
Cubic splines with constant knot spacing | p. 248 |
Knot positions and the calculation of spline approximations | p. 254 |
The distribution of knots at a singularity | p. 254 |
Interpolation for general knots | p. 257 |
The approximation of functions to prescribed accuracy | p. 261 |
The Peano kernel theorem | p. 268 |
The error of a formula for the solution of differential equations | p. 268 |
The Peano kernel theorem | p. 270 |
Application to divided differences and to polynomial interpolation | p. 274 |
Application to cubic spline interpolation | p. 277 |
Natural and perfect splines | p. 283 |
A variational problem | p. 283 |
Properties of natural splines | p. 285 |
Perfect splines | p. 290 |
Optimal interpolation | p. 298 |
The optimal interpolation problem | p. 298 |
L[subscript 1] approximation by B-splines | p. 301 |
Properties of optimal interpolation | p. 307 |
The Haar condition | p. 313 |
Related work and references | p. 317 |
Index | p. 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