Multivariate polysplines: applications to numerical and wavelet analysis

Front cover ......Page 1
Short title ......Page 2
Dedication ......Page 3
Title page ......Page 4
Date-line ......Page 5
Contents ......Page 6
Preface ......Page 14
1 Introduction ......Page 16
1.1.1 Part I: Introduction of polysplines ......Page 18
1.1.2 Part II: Cardinal polysplines ......Page 19
1.1.3 Part III: Wavelet analysis using polysplines ......Page 20
1.2 Audience ......Page 21
1.3 Statements ......Page 22
1.4 Acknowledgements ......Page 23
1.5.1 The operator, object and data concepts of the polyharmonic paradigm ......Page 25
1.5.2 The Taylor formula ......Page 26
Part I Introduction to polysplines ......Page 30
2.1 Cubic splines ......Page 34
2.2 Linear splines ......Page 36
2.3 Variational (Holladay) property of the odd-degree splines ......Page 38
2.4 Existence and uniqueness of odd-degree splines ......Page 41
2.5 The Holladay theorem ......Page 42
3.1 The data concept in two dimensions according to the polyharmonic paradigm ......Page 44
3.1.1 "Parallel lines" or "strips" ......Page 47
3.1.2 "Concentric circles" or "annuli" ......Page 48
3.2.1 The "strips" ......Page 49
3.2.2 The "annuli" ......Page 51
4 The objects concept: harmonic and polyharmonic functions in rectangular domains in $\mathbb{R}^2$ ......Page 54
4.1 Harmonic functions in strips or rectangles ......Page 55
4.2 "Parametrization" of the space of periodic harmonic functions in the strip: the Dirichlet problem ......Page 58
4.3.1 The biharmonic case ......Page 61
4.3.2 The polyharmonic case ......Page 64
4.4 Nonperiodicity in y ......Page 67
5 Polysplines on strips in $\mathbb{R}^2$ ......Page 72
5.1 Periodic harmonic polysplines on strips, $p=1$ ......Page 74
5.2.1 The smoothness scale of the polysplines ......Page 75
5.3 Computing the biharmonic polysplines on strips ......Page 76
5.4 Uniqueness of the interpolation polysplines ......Page 79
6.1 Smoothing airborne magnetic field data ......Page 82
6.2.1 Parallel data lines $\Gamma_j$ ......Page 86
6.2.2 Nonparallel data curves $\Gamma_j$ ......Page 89
6.3 Conclusions ......Page 90
7.1 Harmonic functions in spherical (circular) domains ......Page 92
7.1.1 Harmonic functions in the annulus ......Page 94
7.1.2 "Parametrization" of the space of harmonic functions in the annulus and the ball: the Dirichlet problem ......Page 97
7.1.3 The Dirichlet problem in the ball ......Page 100
7.2 Biharmonic and polyharmonic functions ......Page 101
7.2.1 Polyharmonic functions in annulus and circle ......Page 102
7.2.2 The set of solutions of $L^p_{(k)} u(r)$ = 0 ......Page 104
7.2.3 The operators $L^p_{(k)} (d/dr)$ generate an Extended Complete Chebyshev system ......Page 105
7.3.2 The biharmonic case ......Page 107
7.3.3 The polyharmonic case ......Page 110
7.3.4 Another approach to "parametrization": the Almansi representation ......Page 111
7.3.5 Radially symmetric polyharmonic functions ......Page 112
7.3.6 Another proof of the representation of radially symmetric polyharmonic functions ......Page 113
8 Polysplines on annuli in $\mathbb{R}^2$ ......Page 116
8.1 The biharmonic polysplines, $p = 2$ ......Page 118
8.2 Radially symmetric interpolation polysplines ......Page 119
8.2.1 Applying the change of variable $v = \log r$ ......Page 122
8.2.2 The radially symmetric biharmonic polysplines ......Page 123
8.3 Computing the polysplines for general (nonconstant) data ......Page 124
8.4 The uniqueness of interpolation polysplines on annuli ......Page 125
8.5 The change $v = \log r$ and the operators $M_{k,p}$ ......Page 126
8.6 The fundamental set of solutions for the operator $M_{k,p}(d/dv)$ ......Page 128
9 Polysplines on strips and annuli in $\mathbb{R}^n$ ......Page 132
9.1 Polysplines on strips in $\mathbb{R}^n$ ......Page 133
9.1.1 Polysplines on strips with data periodic in $y$ ......Page 134
9.1.2 Polysplines on strips with compact data ......Page 136
9.2 Polysplines on annuli in $\mathbb{R}^n$ ......Page 137
9.2.1 Biharmonic polysplines in $\mathbb{R}^3$ and $\mathbb{R}^4$ ......Page 142
9.2.2 An "elementary" proof of the existence of interpolation polysplines ......Page 143
10.1 Introduction ......Page 144
10.2 Notations ......Page 145
10.3 Spherical coordinates and the Laplace operator ......Page 146
10.4 Fourier series and basic properties ......Page 149
10.5.2 The functions $r^k\cos k\phi$ and $r^k\sin k\phi$ are polynomials ......Page 151
10.5.4 The functions $r^k\cos k\phi$ and $r^k\sin k\phi$ are a basis of the homogeneous harmonic polynomials of degree $k$ ......Page 152
10.6 Homogeneous polynomials in $\mathbb{R}^n$ ......Page 153
10.7 Gauss representation of homogeneous polynomials ......Page 154
10.7.1 Gauss representation in $\mathbb{R}^2$ ......Page 155
10.7.2 Gauss representation in $\mathbb{R}^n$ ......Page 156
10.8 Gauss representation: analog to the Taylor series, the polyharmonic paradigm ......Page 160
10.8.1 The Almansi representation ......Page 161
10.9 The sets $\mathcal{H}_k$ are eigenspaces for the operator $\Delta_\theta$ ......Page 162
10.10 Completeness of the spherical harmonics in $L_2(\mathbb{S}^{n-1})$ ......Page 164
10.11 Solutions of $\Delta w(x)=0$ with separated variables ......Page 167
10.12 Zonal harmonics $Z_{\theta'}^{(k)}(\theta)$: the functional approach ......Page 168
10.13 The classical approach to zonal harmonics ......Page 174
10.14 The representation of polyharmonic functions using spherical harmonics ......Page 179
10.14.1 Representation of harmonic functions using spherical harmonics ......Page 181
10.14.3 Operator with constant coefficients equivalent to the spherical operator $L^p_{(k)}$ ......Page 183
10.14.4 Representation of polyharmonic functions in annulus and ball ......Page 188
10.15 The operator $r^{n-1} L^p_{(k)}$ is formally self-adjoint ......Page 192
10.16 The Almansi theorem ......Page 194
10.17 Bibliographical notes ......Page 200
11.1 Differential operators and Extended Complete Chebyshev systems ......Page 202
11.2.1 The classical polynomial case ......Page 206
11.2.2 Divided difference operators for Chebyshev systems ......Page 209
11.2.3 Lagrange-Hermite interpolation formula for Chebyshev systems ......Page 211
11.3 Dual operator and ECT-system ......Page 212
11.3.1 Green's function and Taylor formula ......Page 213
11.4 Chebyshev splines and one-sided basis ......Page 214
11.4.1 $TB$-splines, or the Chebyshev $B$-splines as a Peano kernel for the divided difference ......Page 216
11.4.2 Dual basis and Riesz basis property for the $TB$-splines ......Page 218
11.5 Natural Chebyshev splines ......Page 219
12 Appendix on Fourier series and Fourier transform ......Page 224
12.1 Bibliographical notes ......Page 227
Bibliography to Part I ......Page 228
Part II Cardinal polysplines in $\mathbb{R}^n$ ......Page 232
13.1 Cardinal $L$-splines and the interpolation problem ......Page 236
13.2 Differential operators and their solution sets $U_{Z+1}$ ......Page 241
13.3 Variation of the set $U_{Z+1} [\Lambda]$ with $\Lambda$ and other properties ......Page 243
13.4 The Green function $\phi_Z^+(x) of the operator $\mathcal{L}_{Z+1}$ ......Page 244
13.6 The generalized Euler polynomials $A_Z(x; \lambda)$ ......Page 247
13.7 Generalized divided difference operator ......Page 251
13.8 Zeros of the Euler-Frobenius polynomial $\Pi_Z(\lambda)$ ......Page 252
13.9 The cardinal interpolation problem for $L$-splines ......Page 253
13.10 The cardinal compactly supported $L$-splines $Q_{Z+1}$ ......Page 254
13.11 Laplace and Fourier transform of the cardinal $TB$-spline $Q_{Z+1}$ ......Page 256
13.12 Convolution formula for cardinal $TB$-splines ......Page 258
13.13 Differentiation of cardinal $TB$-splines ......Page 259
13.14 Hermite-Gennocchi-type formula ......Page 260
13.15 Recurrence relation for the $TB$-spline ......Page 261
13.16 The adjoint operator $\mathcal{L}_{Z+1}^\ast$ and the $TB$-spline $Q_{Z+1}^\ast(x)$ ......Page 263
13.17 The Euler polynomial $A_Z(x; \lambda)$ and the $TB$-spline $Q_{Z+1}(x)$ ......Page 265
13.18 The leading coefficient of the Euler-Frobenius polynomial $\Pi_Z(\lambda)$ ......Page 268
13.19 Schoenberg's "exponential" Euler $L$-spline $\Phi_Z(x; \lambda)$ and $A_Z(x; \lambda)$ ......Page 269
13.21 Peano kernel and the divided difference operator in the cardinal case ......Page 272
13.22 Two-scale relation (refinement equation) for the $TB$-splines $Q_{Z+1} [\Lambda; h]$ ......Page 274
13.23 Symmetry of the zeros of the Euler-Frobenius polynomial $\Pi_Z(\lambda)$ ......Page 276
13.24 Estimates of the functions $A_Z(x; \lambda)$ and $Q_{Z+1}(x)$ ......Page 279
14 Riesz bounds for the cardinal $L$-splines $Q_{Z+1}$ ......Page 282
14.1 Summary of necessary results for cardinal $L$-splines ......Page 285
14.2 Riesz bounds ......Page 286
14.3 The asymptotic of $A_Z(0; \lambda)$ in $k$ ......Page 293
14.4 Asymptotic of the Riesz bounds $A$, $B$ ......Page 296
14.4.1 Asymptotic for $TB$-splines $Q_{Z+1}$ on the mesh $h\mathbb{Z}$ ......Page 297
14.5 Synthesis of compactly supported polysplines on annuli ......Page 298
15.1 Introduction ......Page 302
15.2 Formulation of the cardinal interpolation problem for polysplines ......Page 303
15.3 $\alpha = 0$ is good for all $L$-splines with $L = M_{k,p}$ ......Page 305
15.4 Explaining the problem ......Page 308
15.5 Schoenberg's results on the fundamental spline $L(X)$ in the polynomial case ......Page 309
15.6 Asymptotic of the zeros of $\Pi_Z(\lambda; 0)$ ......Page 313
15.7 The fundamental spline function $L(X)$ for the spherical operators $M_{k,p}$ ......Page 315
15.7.1 Estimate of the fundamental spline $L(X)$ ......Page 318
15.7.2 Estimate of the cardinal spline $S(X)$ ......Page 319
15.8 Synthesis of the interpolation cardinal polyspline ......Page 320
15.9 Bibliographical notes ......Page 321
Bibliography to Part II ......Page 322
Part III Wavelet analysis ......Page 324
16.1 Cardinal splines and the sets $V_j$ ......Page 328
16.2 The wavelet spaces $W_j$ ......Page 330
16.3 The mother wavelet $\psi$ ......Page 332
16.4 The dual mother wavelet $\tilde\psi$ ......Page 333
16.6 Decomposition relations ......Page 334
16.7 Decomposition and reconstruction algorithms ......Page 336
16.8 Zero moments ......Page 337
16.9 Symmetry and asymmetry ......Page 338
17 Cardinal L-spline wavelet analysis ......Page 340
17.1 Introduction: the spaces $V_j$ and $W_j$ ......Page 341
17.2 Multiresolution analysis using $L$-splines ......Page 344
17.3 The two-scale relation for the $TB$-splines $Q_{Z+1}(x)$ ......Page 346
17.4 Construction of the mother wavelet $\psi_h$ ......Page 348
17.5 Some algebra of Laurent polynomials and the mother wavelet $\psi_h$ ......Page 352
17.6 Some algebraic identities ......Page 354
17.7 The function $\psi_h$ generates a Riesz basis of $W_0$ ......Page 358
17.8 Riesz basis from all wavelet functions $\psi_{2^{-j}h} (x)$ ......Page 360
17.9 The decomposition relations for the scaling function $Q_{Z+1}$ ......Page 367
17.10 The dual scaling function $\tilde\phi$ and the dual wavelet $\tilde\psi$ ......Page 371
17.11 Decomposition and reconstruction by $L$-spline wavelets and MRA ......Page 377
17.12 Discussion of the standard scheme of MRA ......Page 383
18 Polyharmonic wavelet analysis: scaling and rotationally invariant spaces ......Page 386
18.1 The refinement equation for the normed $TB$-spline $Q_{Z+1}$ ......Page 387
18.2 Finding the way: some heuristics ......Page 388
18.3 The sets $PV_j$ and isomorphisms ......Page 390
18.4 Spherical Riesz basis and father wavelet ......Page 392
18.5 Polyharmonic MRA ......Page 394
18.6 Decomposition and reconstruction for polyharmonic wavelets and the mother wavelet ......Page 399
18.7 Zero moments of polyharmonic wavelets ......Page 406
18.8 Bibliographical notes ......Page 408
Bibliography to Part III ......Page 410
Part IV Polysplines for general interfaces ......Page 412
19.1 Introduction ......Page 414
19.2 The setting of the variational problem ......Page 416
19.3 Polysplines of arbitrary order $p$ ......Page 418
19.4 Counting the parameters ......Page 419
19.5 Main results and techniques ......Page 420
19.6 Open problems ......Page 421
20.1 Introduction ......Page 424
20.2 Definition of polysplines ......Page 426
20.3 Basic identity for polysplines of even order $p = 2q$ ......Page 430
20.3.1 Identity for $L=\Delta^{2q}$ ......Page 431
20.3.2 Identity for the operator $L=L_1^2$ ......Page 432
20.4 Uniqueness of interpolation polysplines and extremal Holladay-type property ......Page 436
20.4.1 Holladay property ......Page 440
21.1 Basic proposition for interface on the real line ......Page 444
21.2 A priori estimates in a bounded domain with interfaces ......Page 447
21.3 Fredholm operator in the space $H^{2p+r} (D\\ST) for $r\geq 0$ ......Page 451
21.3.1 The space $\Lambda_1$ for $L=\Delta^p$ ......Page 452
21.3.3 The set $\Lambda_1$ for general elliptic operator $L$ ......Page 457
22.1 Polysplines of order $2q$ for operator $L=L_1^2$ ......Page 460
22.2 The case of a general operator $L$ ......Page 462
22.3 Existence of polysplines on strips with compact data ......Page 465
22.4 Classical smoothness of the interpolation data $g_j$ ......Page 466
22.5 Sobolev embedding in $C^{k,\alpha}$ ......Page 467
22.6 Existence for an interface which is not $C^\infty$ ......Page 468
22.7 Convergence properties of the polysplines ......Page 469
22.8 Bibliographical notes and remarks ......Page 474
23.1.1 Sobolev spaces on manifolds without boundary ......Page 476
23.1.2 Sobolev spaces on the torus $\mathbb{T}^n$ ......Page 478
23.1.3 Sobolev spaces on the sphere $\mathbb{S}^{n-1}$ ......Page 479
23.1.4 Hoelder spaces ......Page 480
23.1.5 Sobolev spaces on manifolds with boundary ......Page 481
23.1.6 Uniform $C^m$ -regularity of $\partial\Omega$ ......Page 482
23.1.7 Trace theorem ......Page 483
23.1.8 The general Sobolev-type embedding theorems ......Page 484
23.1.9 Smoothness across interfaces ......Page 485
23.2 Regular elliptic boundary value problems ......Page 487
23.2.1 Regular elliptic boundary value problems in $\mathbb{R}^n_+$ ......Page 489
23.3 Boundary operators, adjoint problem and Green formula ......Page 490
23.3.1 Boundary operators in neighboring domains ......Page 492
23.4 Elliptic boundary value problems ......Page 494
23.4.2 Fredholm operator ......Page 495
23.4.3 Elliptic boundary value problems in Hoelder spaces ......Page 497
23.4.4 Schauder's continuous parameter method ......Page 498
23.5 Bibliographical notes ......Page 499
24 Afterword ......Page 500
Bibliography to Part IV ......Page 502
Index ......Page 506