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.
Author(s): Marek Kowalski, Christopher Sikorski, Frank Stenger
Series: International Series of Monographs on Computer Science
Edition: First Edition and First Printing
Publisher: Oxford University Press
Year: 1995
Language: English
Pages: 366
City: New York
Contents......Page 12
1.1 General results......Page 18
1.1.1 Exercises......Page 29
1.2 Approximation in unitary spaces......Page 30
1.2.1 Computing the best approximation......Page 34
1.2.2 Completeness of orthogonal systems......Page 37
1.2.3 Examples of orthogonal systems......Page 38
1.2.4 Remarks on convergence of Fourier series......Page 51
1.2.5 Exercises......Page 53
1.3 Uniform approximation......Page 56
1.3.1 Chebyshev subspaces......Page 59
1.3.2 Maximal functionals......Page 64
1.3.3 The Remez algorithm......Page 73
1.3.4 The Korovkin operators......Page 75
1.3.5 Quality of polynomial approximations......Page 80
1.3.6 Converse theorems in polynomial approximation......Page 83
1.3.7 Projection operators......Page 89
1.3.8 Exercises......Page 100
1.4 Annotations......Page 104
1.5 References......Page 106
2.1 Polynomial splines......Page 110
2.1.1 Exercises......Page 119
2.2 B-splines......Page 120
2.2.1 General spline interpolation......Page 126
2.2.2 Exercises......Page 127
2.3 General splines......Page 128
2.4 Annotations......Page 131
2.5 References......Page 132
3.1 Basic definitions......Page 134
3.1.1 Exercises......Page 142
3.2 Interpolation and quadrature......Page 143
3.2.1 Exercises......Page 149
3.3 Approximation of derivatives on Γ......Page 151
3.4 Sinc indefinite integral over Γ......Page 153
3.5 Sinc indefinite convolution over Γ......Page 156
3.5.1 Derivation and justification of procedure......Page 158
3.5.2 Multidimensional indefinite convolutions......Page 163
3.5.3 Two dimensional convolution......Page 164
3.5.4 Exercises......Page 166
3.7 References......Page 167
4.1 Positive base approximation......Page 170
4.2 Approximation via elliptic functions......Page 175
4.2.1 Exercises......Page 177
4.3 Heaviside, filter, and delta functions......Page 178
4.3.1 Heaviside function......Page 179
4.3.2 The filter or characteristic function......Page 180
4.3.3 The impulse or delta function......Page 181
4.5 References......Page 183
5 Moment Problems......Page 186
5.1 Duality with approximation......Page 187
5.2 The moment problem in the space C[sub(o)](D)......Page 192
5.3 Classical moment problems......Page 195
5.3.1 Exercises......Page 202
5.4 Density and determinateness......Page 206
5.4.1 Exercises......Page 220
5.5 A Sinc moment problem......Page 222
5.6 Multivariate orthogonal polynomials......Page 223
5.6.1 Exercises......Page 235
5.7 Annotations......Page 236
5.8 References......Page 237
6.1 n-Widths......Page 240
6.1.1 Relationships between n-widths......Page 246
6.1.2 Algebraic versions of a[sub(n)] and c[sub(n)]......Page 252
6.1.3 Exercises......Page 253
6.2 s-Numbers......Page 254
6.2.1 s-Numbers and singular values......Page 257
6.2.2 Relationships between s-numbers......Page 263
6.3 Annotations......Page 272
6.4 References......Page 273
7 Optimal Approximation Methods......Page 276
7.1 A general approximation problem......Page 279
7.1.1 Radius of information—optimal algorithms......Page 281
7.2 Linear problems......Page 287
7.2.1 Optimal information......Page 293
7.2.2 Relations to n-widths......Page 298
7.2.3 Exercises......Page 302
7.3 Parallel versus sequential methods......Page 303
7.3.1 Exercises......Page 307
7.4 Linear and spline algorithms......Page 308
7.4.1 Spline algorithms......Page 312
7.4.2 Relations to linear Kolmogorov n-widths......Page 319
7.5 s-Numbers, minimal errors......Page 321
7.5.1 Exercises......Page 326
7.6 Optimal methods......Page 327
7.6.1 Optimal complexity methods for linear problems......Page 329
7.7 Annotations......Page 331
7.8 References......Page 333
8.1 Sinc solution of Burgers' equation......Page 336
8.2.1 Formulation of the problem......Page 338
8.2.2 Relations to n-widths......Page 339
8.2.3 Algorithms and their errors......Page 342
8.2.4 Asymptotics of minimal cost......Page 349
8.2.5 Exercises......Page 350
8.3.1 Formulation of the problem......Page 351
8.3.2 Optimality theorem......Page 352
8.5 References......Page 357
B......Page 360
E......Page 361
I......Page 362
M......Page 363
R......Page 364
T......Page 365
Z......Page 366
