| 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. |
