Approximation Theory and Methods

ISBN-10: 0521295149

ISBN-13: 9780521295147

Edition: 1981

Authors: M. J. D. Powell
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Description: 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.

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Book details

Copyright year: 1981
Publisher: Cambridge University Press
Publication date: 3/31/1981
Binding: Paperback
Pages: 352
Size: 6.00" wide x 9.00" long x 0.75" tall
Weight: 1.034
Language: English

Preface
The approximation problem and existence of best approximations
Examples of approximation problems
Approximation in a metric space
Approximation in a normed linear space
The L[subscript p]-norms
A geometric view of best approximations
The uniqueness of best approximations
Convexity conditions
Conditions for the uniqueness of the best approximation
The continuity of best approximation operators
The 1-, 2- and [infinity]-norms
Approximation operators and some approximating functions
Approximation operators
Lebesgue constants
Polynomial approximations to differentiable functions
Piecewise polynomial approximations
Polynomial interpolation
The Lagrange interpolation formula
The error in polynomial interpolation
The Chebyshev interpolation points
The norm of the Lagrange interpolation operator
Divided differences
Basic properties of divided differences
Newton's interpolation method
The recurrence relation for divided differences
Discussion of formulae for polynomial interpolation
Hermite interpolation
The uniform convergence of polynomial approximations
The Weierstrass theorem
Monotone operators
The Bernstein operator
The derivatives of the Bernstein approximations
The theory of minimax approximation
Introduction to minimax approximation
The reduction of the error of a trial approximation
The characterization theorem and the Haar condition
Uniqueness and bounds on the minimax error
The exchange algorithm
Summary of the exchange algorithm
Adjustment of the reference
An example of the iterations of the exchange algorithm
Applications of Chebyshev polynomials to minimax approximation
Minimax approximation on a discrete point set
The convergence of the exchange algorithm
The increase in the levelled reference error
Proof of convergence
Properties of the point that is brought into reference
Second-order convergence
Rational approximation by the exchange algorithm
Best minimax rational approximation
The best approximation on a reference
Some convergence properties of the exchange algorithm
Methods based on linear programming
Least squares approximation
The general form of a linear least squares calculation
The least squares characterization theorem
Methods of calculation
The recurrence relation for orthogonal polynomials
Properties of orthogonal polynomials
Elementary properties
Gaussian quadrature
The characterization of orthogonal polynomials
The operator R[subscript n]
Approximation to periodic functions
Trigonometric polynomials
The Fourier series operator S[subscript n]
The discrete Fourier series operator
Fast Fourier transforms
The theory of best L[subscript 1] approximation
Introduction to best L[subscript 1] approximation
The characterization theorem
Consequences of the Haar condition
The L[subscript 1] interpolation points for algebraic polynomials
An example of L[subscript 1] approximation and the discrete case
A useful example of L[subscript 1] approximation
Jackson's first theorem
Discrete L[subscript 1] approximation
Linear programming methods
The order of convergence of polynomial approximations
Approximations to non-differentiable functions
The Dini-Lipschitz theorem
Some bounds that depend on higher derivatives
Extensions to algebraic polynomials
The uniform boundedness theorem
Preliminary results
Tests for uniform convergence
Application to trigonometric polynomials
Application to algebraic polynomials
Interpolation by piecewise polynomials
Local interpolation methods
Cubic spline interpolation
End conditions for cubic spline interpolation
Interpolating splines of other degrees
B-splines
The parameters of a spline function
The form of B-splines
B-splines as basis functions
A recurrence relation for B-splines
The Schoenberg-Whitney theorem
Convergence properties of spline approximations
Uniform convergence
The order of convergence when f is differentiable
Local spline interpolation
Cubic splines with constant knot spacing
Knot positions and the calculation of spline approximations
The distribution of knots at a singularity
Interpolation for general knots
The approximation of functions to prescribed accuracy
The Peano kernel theorem
The error of a formula for the solution of differential equations
The Peano kernel theorem
Application to divided differences and to polynomial interpolation
Application to cubic spline interpolation
Natural and perfect splines
A variational problem
Properties of natural splines
Perfect splines
Optimal interpolation
The optimal interpolation problem
L[subscript 1] approximation by B-splines
Properties of optimal interpolation
The Haar condition
Related work and references
Index
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