Approximate approximations

Title page ......Page 1
Abstract ......Page 2
Contents ......Page 3
Preface ......Page 9
1.1.1. Exercise for a freshman ......Page 13
1.1.2. Simple approximation formula ......Page 16
1.2.1. Errors of approximate quasi-interpolation ......Page 18
1.2.2. A simple application of the approximation formula (1.7) ......Page 20
1.2.3. Other basis functions ......Page 22
1.2.4. Examples of higher-order quasi-interpolants ......Page 25
1.2.5. Examples of multi-dimensional quasi-interpolants ......Page 29
2.1.1. Function spaces ......Page 31
2.1.2. Fourier transform ......Page 32
2.1.3. Convolutions ......Page 33
2.1.5. Multi-dimensional Poisson summation formula ......Page 34
2.2.1. Young\'s inequality for semi-discrete convolutions ......Page 35
2.2.2. An auxiliary function ......Page 37
2.2.3. Representations of the quasi-interpolant $\\mathcal{M}_{h,\\mathcal{D}}$ ......Page 39
2.2.4. Relations to continuous convolutions ......Page 41
2.2.5. Poisson\'s summation formula for $\\sigma_\\alpha$ ......Page 43
2.3.1. Using the moments of $\\eta$ ......Page 45
2.3.2. Truncation of summation ......Page 47
2.3.3. Local estimate of the quasi-interpolation ......Page 48
2.3.4. Holder continuous functions ......Page 50
2.3.5. Approximation of derivatives ......Page 52
2.4.1. Formulation of the result ......Page 53
2.4.2. Estimation of the remainder term ......Page 54
2.4.3. Remark on the best approximation ......Page 56
2.5. Notes ......Page 57
3.2.1. One-dimensional examples ......Page 61
3.3.1. A general formula ......Page 62
3.3.2. Symmetric basis functions ......Page 63
3.3.3. Radial basis functions ......Page 64
3.3.4. Example ......Page 66
3.3.5. Anisotropic Gaussian function ......Page 68
3.4. Some other methods ......Page 69
3.4.1. Using a collection of different $\\mathcal{D}$ ......Page 70
3.4.2. Derivatives with respect to $\\mathcal{D}$ ......Page 71
3.5.1. Construction of the generating function (3.30) ......Page 72
3.5.2. The case of symmetric $\\eta$ ......Page 75
3.6. Matrix-valued basis functions ......Page 76
3.7.1. Perturbation of $\\eta$ ......Page 77
3.7.2. Combinations of translates ......Page 79
3.8. Notes ......Page 80
4.1. Introduction ......Page 81
4.2. Hilbert transform ......Page 83
4.3.1. Action on Gaussians ......Page 86
4.3.2. Action on higher-order basis functions ......Page 88
4.3.3. Semi-analytic cubature formulas ......Page 89
4.3.4. Numerical example ......Page 90
4.3.5. Gradient of the harmonic potential ......Page 91
4.4.1. Rough error estimate ......Page 93
4.4.2. Quasi-interpolation error in Sobolev spaces of negative order ......Page 94
4.4.3. Cubature error for the harmonic potential ......Page 97
4.5. Application to boundary integral equation methods ......Page 98
4.5.1. Transformation of inhomogeneous problems ......Page 99
4.5.2. Error estimates ......Page 100
4.6. Notes ......Page 103
5.1. Diffraction potentials ......Page 105
5.1.2. Action of the one-dimensional diffraction potential on the Gaussian ......Page 106
5.1.4. Analytic expressions of the diffraction potential ......Page 108
5.2. Potentials of advection-diffusion operators ......Page 110
5.3. Elastic and hydrodynamic potentials ......Page 111
5.4.2. An auxiliary formula ......Page 112
5.4.3. Potentials of the Gaussian ......Page 113
5.4.4. Potentials of higher-order generating functions ......Page 114
5.4.5. Final form of the elastic and hydrodynamic potentials ......Page 117
5.4.6. Using matrix-valued basis functions ......Page 118
5.5. Three-dimensional potentials ......Page 120
5.5.1. Potentials of the Gaussian ......Page 121
5.5.2. Elastic potential of higher-order generating functions ......Page 122
5.5.3. Final form of the elastic and hydrodynamic potentials ......Page 124
5.6. Notes ......Page 125
6.1.1. Square root of the Laplacian ......Page 127
6.1.2. Higher-dimensional singular integrals ......Page 128
6.1.3. Biharmonic potential ......Page 131
6.2.1. The Cauchy problem for the heat equation ......Page 132
6.2.2. The Cauchy problem for the wave equation ......Page 135
6.2.3. Vibrations of a plate ......Page 137
6.3. Potentials of anisotropic Gaussians ......Page 138
6.3.1. Second-order problems ......Page 139
6.3.2. Some special cases ......Page 140
6.3.3. Elastic and hydrodynamic potentials of anisotropic Gaussians ......Page 143
6.3.4. Potentials of orthotropic Gaussians ......Page 145
6.4. Potentials of higher-order generating functions for orthotropic Gaussians ......Page 146
6.5.1. Integral operators over bounded domains ......Page 148
6.5.2. Integral operators over the half-space ......Page 149
6.5.3. Harmonic potentials ......Page 150
6.5.4. Diffraction potentials ......Page 151
6.6. Notes ......Page 153
7.1.1. Approximants on coarse grids ......Page 155
7.1.2. Examples ......Page 157
7.2.1. Generating functions ......Page 159
7.2.2. Saturation error ......Page 160
7.3.1. Lagrangian function ......Page 163
7.3.2. Simplification ......Page 166
7.3.3. Interpolation error ......Page 168
7.3.4. Spectral convergence ......Page 169
7.3.5. Exponential convergence ......Page 172
7.4. Orthogonal projection ......Page 174
7.5. Notes ......Page 175
8.1. Introduction ......Page 177
8.2.1. Approximate refinement equations for $\\eta$ ......Page 180
8.2.2. Examples of mask functions ......Page 182
8.3.1. Approximate chain of subspaces ......Page 183
8.3.2. Almost orthogonal decomposition ......Page 184
8.4.1. Gaussian as scaling function ......Page 185
8.4.2. Moments ......Page 187
8.4.3. Other examples ......Page 189
8.5.1. Projections onto $\\mathbf{V}_0$ ......Page 193
8.5.2. Projections onto $\\mathbf{W}_0$ ......Page 194
8.6. Proof of Theorem 8.6 ......Page 195
8.6.2. Error of replacing $G(\\lambda)$ by $g(\\lambda)$ ......Page 196
8.6.3. Equations for the Fourier coefficients of $1/g$ ......Page 197
8.6.4. Solution of (8.58) ......Page 200
8.6.5. Error of replacing $a_k$ by $\\tilde{a}_k$ ......Page 201
8.7.1. Representation of wavelet basis functions ......Page 202
8.7.2. Potentials of anisotropic Gaussians of a complex argument ......Page 204
8.7.3. Harmonic potentials of approximate wavelets ......Page 205
8.7.5. Elastic and hydrodynamic potentials of approximate wavelets ......Page 206
8.8. Numerical example ......Page 207
8.9. Notes ......Page 208
9.1. Introduction ......Page 209
9.2. Simple approach ......Page 210
9.3.1. Approximate factorization of the quasi-interpolant ......Page 212
9.3.2. Properties of the mask function $\\tilde{\\eta}$ ......Page 213
9.3.3. Convolutions based on the remainder term ......Page 214
9.4.1. Multi-resolution decomposition ......Page 215
9.4.2. Restriction of $\\mathcal{A}_j$ ......Page 216
9.4.3. Pointwise estimates ......Page 217
9.4.4. Numerical examples ......Page 219
9.4.5. $L_p$-estimates for quasi-interpolants of functions in domains ......Page 222
9.4.7. Estimates in weak norms ......Page 225
9.5. Cubature of potentials in domains ......Page 226
9.6. Anisotropic boundary layer approximate approximation ......Page 230
9.7. Potentials of tensor product generating functions ......Page 233
9.8. Notes ......Page 235
10.1.1. Orthotropic grid ......Page 237
10.1.2. Non-orthogonal grid ......Page 238
10.1.3. Examples ......Page 239
10.2.1. Perturbed grid ......Page 242
10.2.3. Error estimates ......Page 243
10.2.4. Numerical experiments with quasi-interpolants ......Page 245
10.3.1. Description of construction and error estimate ......Page 247
10.3.2. Quasi-interpolation on domains ......Page 249
10.3.3. Quasi-interpolation on manifolds ......Page 252
10.3.4. Quasi-interpolant of general form ......Page 253
10.4. Notes ......Page 255
Chapter 11. Scattered data approximate approximations ......Page 257
11.1.1. Basis functions with compact support ......Page 258
11.1.2. Basis functions with non-compact support ......Page 259
11.2. Quasi-interpolants of a general form ......Page 261
11.2.1. Compactly supported basis functions ......Page 262
11.2.2. Quasi-interpolants with non-compactly supported basis functions ......Page 263
11.3. Computation of integral operators ......Page 264
11.4.1. Approximate partition of unity on a piecewise uniform grid ......Page 266
11.4.2. Scattered nodes close to a piecewise uniform grid ......Page 268
11.4.3. Discrepancy as convolution ......Page 269
11.4.4. Construction of polynomials ......Page 272
11.4.5. Existence and uniqueness ......Page 273
11.4.6. Approximation error in the case $d_j=\\mathcal{D}$ ......Page 279
11.4.7. Approximation error in the case $d_j<\\mathcal{D}$ ......Page 280
11.4.8. Summary ......Page 285
11.4.9. Numerical experiments ......Page 288
11.5. Notes ......Page 290
Chapter 12. Numerical algorithms based upon approximate approximations — linear problems ......Page 293
12.1.1. Problem ......Page 294
12.1.2. Error analysis for the collocation method ......Page 297
12.1.3. Numerical example ......Page 300
12.2.1. Boundary integral equations ......Page 301
12.2.2. Boundary point method ......Page 304
12.2.3. BPM for single layer potentials ......Page 306
12.2.4. BPM for double layer potentials ......Page 307
12.2.5. Numerical experiments ......Page 309
12.2.6. Stability analysis ......Page 312
12.3. Computation of multi-dimensional single layer harmonic potentials ......Page 318
12.3.1. Asymptotic formulas ......Page 321
12.3.2. Approximation error ......Page 325
12.3.3. Cubature formula ......Page 326
12.3.4. Basis functions ......Page 329
12.3.5. Numerical examples ......Page 331
12.4. Notes ......Page 332
13.1.1. One-step time-marching algorithms ......Page 333
13.1.2. Higher-dimensional case ......Page 336
13.2. Application to the non-stationary Navier-Stokes equations ......Page 339
13.2.1. Integral equation formulation ......Page 340
13.2.2. One-step time-marching algorithm ......Page 342
13.2.3. Formulas for the plane problem ......Page 343
13.2.5. Formulas for $\\mathbb{R}_n$, $n>2$ ......Page 345
13.3. Non-local evolution equations ......Page 351
13.3.2. Sivashinsky equation ......Page 352
13.4. Notes ......Page 353
Bibliography ......Page 357
Index ......Page 361