Selected Topics in Approximation and Computation

by Kowalski, Marek; Sikorski, Christopher; Stenger, Frank

• ISBN13: 9780195080599
• ISBN10: 0195080599
• Format: Hardcover
• Copyright: 1995-08-31
• Publisher: Oxford University Press

Summary

Selected Topics in Approximation and Computation addresses the relationship between modern approximation theory and computational methods. The text is a combination of expositions of basic classical methods of approximation leading to popular splines and new explicit tools of computation, including Sinc methods, elliptic function methods, and positive operator approximation methods. It also provides an excellent summary of worst case analysis in information based complexity. It relates optimal computational methods with the theory of s-numbers and n-widths. It can serve as a text for senior-graduate courses in computer science and applied mathematics, and also as a reference for professionals.

Table of Contents

Classical Approximation	p. 1
General results	p. 1
Exercises	p. 12
Approximation in unitary spaces	p. 13
Computing the best approximation	p. 17
Completeness of orthogonal systems	p. 20
Examples of orthogonal systems	p. 21
Remarks on convergence of Fourier series	p. 34
Exercises	p. 36
Uniform approximation	p. 39
Chebyshev subspaces	p. 42
Maximal functionals	p. 47
The Remez algorithm	p. 56
The Korovkin operators	p. 58
Quality of polynomial approximations	p. 63
Converse theorems in polynomial approximation	p. 66
Projection operators	p. 72
Exercises	p. 83
Annotations	p. 87
References	p. 89
Splines	p. 93
Polynomial splines	p. 93
Exercises	p. 102
B-splines	p. 103
General spline interpolation	p. 109
Exercises	p. 110
General splines	p. 111
Exercises	p. 114
Annotations	p. 114
References	p. 115
Sinc Approximation	p. 117
Basic definitions	p. 117
Exercises	p. 125
Interpolation and quadrature	p. 126
Exercises	p. 132
Approximation of derivatives on [Gamma]	p. 134
Exercises	p. 136
Sinc indefinite integral over [Gamma]	p. 136
Exercises	p. 139
Sinc indefinite convolution over [Gamma]	p. 139
Derivation and justification of procedure	p. 141
Multidimensional indefinite convolutions	p. 146
Two dimensional convolution	p. 147
Exercises	p. 149
Annotations	p. 150
References	p. 150
Explicit Sinc-Like Methods	p. 153
Positive base approximation	p. 153
Exercises	p. 158
Approximation via elliptic functions	p. 158
Exercises	p. 160
Heaviside, filter, and delta functions	p. 161
Heaviside function	p. 162
The filter or characteristic function	p. 163
The impulse or delta function	p. 164
Exercises	p. 166
Annotations	p. 166
References	p. 166
Moment Problems	p. 169
Duality with approximation	p. 170
Exercises	p. 175
The moment problem in the space C[subscript o](D)	p. 175
Classical moment problems	p. 178
Exercises	p. 185
Density and determinateness	p. 189
Exercises	p. 203
A Sinc moment problem	p. 205
Exercises	p. 206
Multivariate orthogonal polynomials	p. 206
Exercises	p. 218
Annotations	p. 219
References	p. 220
n-Widths and s-Numbers	p. 223
n-Widths	p. 223
Relationships between n-widths	p. 229
Algebraic versions of a[subscript n] and c[subscript n]	p. 235
Exercises	p. 236
s-Numbers	p. 237
s-Numbers and singular values	p. 240
Relationships between s-numbers	p. 246
Exercises	p. 255
Annotations	p. 255
References	p. 256
Optimal Approximation Methods	p. 259
A general approximation problem	p. 262
Radius of information--optimal algorithms	p. 264
Exercises	p. 270
Linear problems	p. 270
Optimal information	p. 276
Relations to n-widths	p. 281
Exercises	p. 285
Parallel versus sequential methods	p. 286
Exercises	p. 290
Linear and spline algorithms	p. 291
Spline algorithms	p. 295
Relations to linear Kolmogorov n-widths	p. 302
Exercises	p. 304
s-Numbers, minimal errors	p. 304
Exercises	p. 309
Optimal methods	p. 310
Optimal complexity methods for linear problems	p. 312
Exercises	p. 314
Annotations	p. 314
References	p. 316
Applications	p. 319
Sinc solution of Burgers' equation	p. 319
Signal recovery	p. 321
Formulation of the problem	p. 321
Relations to n-widths	p. 322
Algorithms and their errors	p. 325
Asymptotics of minimal cost	p. 332
Exercises	p. 333
Bisection method	p. 334
Formulation of the problem	p. 334
Optimality theorem	p. 335
Exercises	p. 340
Annotations	p. 340
References	p. 340
Index	p. 343
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