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