Multivariate polysplines: applications to numerical and wavelet analysis
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
San Diego [u.a.]
Acad. Press
2001
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 498 S. graph. Darst. |
| ISBN: | 0124224903 |
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| 100 | 1 | |a Kunčev, Ognjan I. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Multivariate polysplines |b applications to numerical and wavelet analysis |c Ognyan Kounchev |
| 264 | 1 | |a San Diego [u.a.] |b Acad. Press |c 2001 | |
| 300 | |a XIV, 498 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 4 | |a Differential equations, Elliptic |x Numerical solutions | |
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| adam_text | Contents
Preface xv
1 Introduction 1
1.1 Organization of material 3
1.1.1 Part I: Introduction of polysplines 3
1.1.2 Part II: Cardinal polysplines 4
1.1.3 Part III: Wavelet analysis using polysplines 5
1.1.4 Part IV: Polysplines on general interfaces 6
1.2 Audience 6
1.3 Statements 7
1.4 Acknowledgements 8
1.5 The polyharmonic paradigm 10
1.5.1 The operator, object and data concepts of the
polyharmonic paradigm 10
1.5.2 The Taylor formula 11
Part I Introduction to polysplines 15
2 One dimensional linear and cubic splines 19
2.1 Cubic splines 19
2.2 Linear splines 21
2.3 Variational (Holladay) property of the odd degree splines 23
2.4 Existence and uniqueness of odd degree splines 26
2.5 The Holladay theorem 27
3 The two dimensional case: data and smoothness concepts 29
3.1 The data concept in two dimensions according to the
polyharmonic paradigm 29
3.1.1 Parallel lines or strips 32
3.1.2 Concentric circles or annuli 33
3.2 The smoothness concept according to the
polyharmonic paradigm 34
3.2.1 The strips 34
3.2.2 The annuli 36
v
vi Contents
4 The objects concept: harmonic and polyharmonic functions in
rectangular domains in K2 39
4.1 Harmonic functions in strips or rectangles 40
4.2 Parametrization of the space of periodic
harmonic functions in the strip: the Dirichlet problem 43
4.3 Parametrization of the space of periodic
polyharmonic functions in the strip: the Dirichlet problem 46
4.3.1 The biharmonic case 46
4.3.2 The polyharmonic case 49
4.4 Nonperiodicity in y 52
5 Polysplines on strips in R2 57
5.1 Periodic harmonic polysplines on strips, p = 1 59
5.2 Periodic biharmonic polysplines on strips, p = 2 60
5.2.1 The smoothness scale of the polysplines 60
5.3 Computing the biharmonic polysplines on strips 61
5.4 Uniqueness of the interpolation polysplines 64
6 Application of polysplines to magnetism and CAGD 67
6.1 Smoothing airborne magnetic field data 67
6.2 Applications to computer aided geometric design 71
6.2.1 Parallel data lines Ty 71
6.2.2 Nonparallel data curves Tj 74
6.3 Conclusions 75
7 The objects concept: harmonic and polyharmonic functions in
annuli in R2 77
7.1 Harmonic functions in spherical (circular) domains 77
7.1.1 Harmonic functions in the annulus 79
7.1.2 Parametrization of the space of harmonic
functions in the annulus and the ball:
the Dirichlet problem 82
7.1.3 The Dirichlet problem in the ball 85
7.1.4 An important change of the variable, v = log r 86
7.2 Biharmonic and polyharmonic functions 86
7.2.1 Polyharmonic functions in annulus and circle 87
7.2.2 The set of solutions of Lp{k)u(r) =0 89
7.2.3 The operators L^Ad/dr) generate an Extended
Complete Chebyshev system 90
7.3 Parametrization of the space of polyharmonic functions
in the annulus and ball: the Dirichlet problem 92
7.3.1 The one dimensional case 92
7.3.2 The biharmonic case 92
7.3.3 The polyharmonic case 95
7.3.4 Another approach to parametrization : the
Almansi representation 96
Contents vii
7.3.5 Radially symmetric polyharmonic functions 97
7.3.6 Another proof of the representation of radially
symmetric polyharmonic functions 98
8 Polysplines on annuli in R2 101
8.1 The biharmonic polysplines, p = 2 103
8.2 Radially symmetric interpolation polysplines 104
8.2.1 Applying the change of variable v = log r 107
8.2.2 The radially symmetric biharmonic polysplines 108
8.3 Computing the polysplines for general
(nonconstant) data 109
8.4 The uniqueness of interpolation polysplines on annuli 110
8.5 The change v = log r and the operators M^ p 111
8.6 The fundamental set of solutions for
the operator Mk,p (d/dv) 113
9 Polysplines on strips and annuli in R 117
9.1 Polysplines on strips in W 118
9.1.1 Polysplines on strips with data periodic in y 119
9.1.2 Polysplines on strips with compact data 121
9.1.3 The case/? = 2 122
9.2 Polysplines on annuli in R 122
9.2.1 Biharmonic polysplines in R3 and R4 127
9.2.2 An elementary proof of the existence of
interpolation polysplines 128
10 Compendium on spherical harmonics and
polyharmonic functions 129
10.1 Introduction 129
10.2 Notations 130
10.3 Spherical coordinates and the Laplace operator 131
10.4 Fourier series and basic properties 134
10.5 Finding the point of view 136
10.5.1 The functions rkcoskip and r*sin ^^ are
harmonic for A 0 136
10.5.2 The functions r*cos k p and rksin kip are
polynomials 136
10.5.3 The functions rkcosk(p and rks mk p are
homogeneous of degree k 0 137
10.5.4 The functions rkcos kip and r^ sin kip are
a basis of the homogeneous harmonic polynomials
of degree A 137
10.5.5 The multidimensional Ansatz 138
10.6 Homogeneous polynomials in R 138
10.6.1 Examples of homogeneous polynomials 139
10.7 Gauss representation of homogeneous polynomials 139
10.7.1 Gauss representation in R2 140
10.7.2 Gauss representation in R 141
viii Contents
10.8 Gauss representation: analog to the Taylor series,
the polyharmonic paradigm 145
10.8.1 The Almansi representation 146
10.9 The sets Hk are eigenspaces for the operator A# 147
10.10 Completeness of the spherical harmonics in L2(S 1) 149
10.11 Solutions of A w (x) = 0 with separated variables 152
10.12 Zonal harmonics Z , (0): the functional approach 153
10.12.1 Estimates of the derivatives of K* (0):
Markov Bernstein type inequality 159
10.13 The classical approach to zonal harmonics 159
10.14 The representation of polyharmonic functions using
spherical harmonics 164
10.14.1 Representation of harmonic functions using
spherical harmonics 166
10.14.2 Solutions of the spherical operator Lp{k)f{r) = 0 168
10.14.3 Operator with constant coefficients equivalent to
the spherical operator L^, 168
10.14.4 Representation of polyharmonic functions in
annulus and ball 173
10.15 The operator r ~l Lp(k) is formally self adjoint 177
10.16 The Almansi theorem 179
10.17 Bibliographical notes 185
11 Appendix on Chebyshev splines 187
11.1 Differential operators and Extended Complete
Chebyshev systems 187
11.2 Divided differences for Extended Complete
Chebyshev systems 191
11.2.1 The classical polynomial case 191
11.2.2 Divided difference operators for
Chebyshev systems 194
11.2.3 Lagrange Hermite interpolation formula for
Chebyshev systems 196
11.3 Dual operator and ECT system 197
11.3.1 Green s function and Taylor formula 198
11.4 Chebyshev splines and one sided basis 199
11.4.1 7 fi splines, or the Chebyshev fi splines as a
Peano kernel for the divided difference 201
11.4.2 Dual basis and Riesz basis property for the
TB splines 203
11.5 Natural Chebyshev splines 204
12 Appendix on Fourier series and Fourier transform 209
12.1 Bibliographical notes 212
Bibliography to Part I 213
Contents ix
Part II Cardinal polysplines in R 217
13 Cardinal L splines according to Micchelli 221
13.1 Cardinal L splines and the interpolation problem 221
13.2 Differential operators and their solution sets Uz+ 226
13.3 Variation of the set Uz+ [A] with A and other properties 228
13.4 The Green function j ^{x) of the operator £z+1 229
13.5 The dictionary: L polynomial case 232
13.6 The generalized Euler polynomials Az(x; X) 232
13.7 Generalized divided difference operator 236
13.8 Zeros of the Euler Frobenius polynomial
nz(X) 237
13.9 The cardinal interpolation problem for L splines 238
13.10 The cardinal compactly supported L splines Qz+ 239
13.11 Laplace and Fourier transform of the cardinal
TB spline Qz+ 241
13.12 Convolution formula for cardinal Tfi splines 243
13.13 Differentiation of cardinal TB splines 244
13.14 Hermite Gennocchi type formula 245
13.15 Recurrence relation for the r/J spline 246
13.16 The adjoint operator C*z+{ and the 7B spline Qz+i (.*) 248
13.17 The Euler polynomial Az(x; X) and the
TB spline Qz+i(x) 250
13.18 The leading coefficient of the Euler Frobenius polynomial
UZ(X) 253
13.19 Schoenberg s exponential Euler L spline $zU; X)
andAz(x X) 254
13.20 Marsden s identity for cardinal L splines 257
13.21 Peano kernel and the divided difference operator in
the cardinal case 257
13.22 Two scale relation (refinement equation) for the
7 fi splines Qz+ [ A; h] 259
13.23 Symmetry of the zeros of the Euler Frobenius polynomial
Tlz(X) 261
13.24 Estimates of the functions A / (x: X) and Q/.+ (x) 264
14 Riesz bounds for the cardinal L splines Qz+ 267
14.1 Summary of necessary results for cardinal L splines 270
14.2 Riesz bounds 271
14.3 The asymptotic of Az(0: X) in k 278
14.4 Asymptotic of the Riesz bounds A. B 281
14.4.1 Asymptotic for 77? splines Qz+ «n the mesh hi 282
14.5 Synthesis of compactly supported polysplines on annuli 283
x Contents
15 Cardinal interpolation polysplines on annuli 287
15.1 Introduction 287
15.2 Formulation of the cardinal interpolation problem
for polysplines 288
15.3 a = Ois good for all L splines with L = M^.p 290
15.4 Explaining the problem 293
15.5 Schoenberg s results on the fundamental spline L(X)
in the polynomial case 294
15.6 Asymptotic of the zeros of Uz(A.; 0) 298
15.7 The fundamental spline function L(X) for the
spherical operators Mf, p 300
15.7.1 Estimate of the fundamental spline L(x) 303
15.7.2 Estimate of the cardinal spline S(x) 304
15.8 Synthesis of the interpolation cardinal polyspline 305
15.9 Bibliographical notes 306
Bibliography to Part II 307
Part III Wavelet analysis 309
16 Chui s cardinal spline wavelet analysis 313
16.1 Cardinal splines and the sets Vj 313
16.2 The wavelet spaces Wj 315
16.3 The mother wavelet ifr 317
16.4 The dual mother wavelet j/ 318
16.5 The dual scaling function (j 319
16.6 Decomposition relations 319
16.7 Decomposition and reconstruction algorithms 321
16.8 Zero moments 322
16.9 Symmetry and asymmetry 323
17 Cardinal L spline wavelet analysis 325
17.1 Introduction: the spaces Vj and Wj 326
17.2 Multiresolution analysis using L splines 329
17.3 The two scale relation for the 7B splines Qz+1 (x) 331
17.4 Construction of the mother wavelet i/r/, 333
17.5 Some algebra of Laurent polynomials and the
mother wavelet ^/, 337
17.6 Some algebraic identities 339
17.7 The function i/ff, generates a Riesz basis of Wo 343
17.8 Riesz basis from all wavelet functions fc Jh (¦*) 345
17.9 The decomposition relations for the scaling function Qz+1 352
17.10 The dual scaling function p and the dual wavelet r/r 356
17.11 Decomposition and reconstruction by
L spline wavelets and MRA 362
17.12 Discussion of the standard scheme of MRA 368
Contents xi
18 Polyharmonic wavelet analysis: scaling and rotationally
invariant spaces 371
18.1 The refinement equation for the normed Tfi spline Qz+1 372
18.2 Finding the way: some heuristics 373
18.3 The sets PVj and isomorphisms 375
18.4 Spherical Riesz basis and father wavelet 377
18.5 Polyharmonic MRA 379
18.6 Decomposition and reconstruction for polyharmonic
wavelets and the mother wavelet 384
18.7 Zero moments of polyharmonic wavelets 391
18.8 Bibliographical notes 393
Bibliography to Part III 395
Part IV Polysplines for general interfaces 397
19 Heuristic arguments 399
19.1 Introduction 399
19.2 The setting of the variational problem 401
19.3 Polysplines of arbitrary order p 403
19.4 Counting the parameters 404
19.5 Main results and techniques 405
19.6 Open problems 406
20 Definition of polysplines and uniqueness for general interfaces 409
20.1 Introduction 409
20.2 Definition of polysplines 411
20.3 Basic identity for polysplines of even order p = 2q 415
20.3.1 Identity for L = A2 416
20.3.2 Identity for the operator L = L2 417
20.4 Uniqueness of interpolation polysplines and
extremal Holladay type property 421
20.4.1 Holladay property 425
21 A priori estimates and Fredholm operators 429
21.1 Basic proposition for interface on the real line 429
21.2 A priori estimates in a bounded domain with interfaces 432
21.3 Fredholm operator in the space H2i +r (D ST) for /• 0 436
21.3.1 The space A i for L = A 7 437
21.3.2 The case L = A2 442
21.3.3 The set Ai for general elliptic operator L 442
22 Existence and convergence of polysplines 445
22.1 Polysplines of order 2q for operator L — L 445
22.2 The case of a general operator L 447
xii Contents
22.3 Existence of polysplines on strips with compact data 450
22.4 Classical smoothness of the interpolation data gj 451
22.5 Sobolev embedding in CkM 452
22.6 Existence for an interface which is not C00 453
22.7 Convergence properties of the polysplines 454
22.8 Bibliographical notes and remarks 459
23 Appendix on elliptic boundary value problems in
Sobolev and Holder spaces 461
23.1 Sobolev and Holder spaces 461
23.1.1 Sobolev spaces on manifolds without boundary 461
23.1.2 Sobolev spaces on the torus T 463
23.1.3 Sobolev spaces on the sphere S ~ 464
23.1.4 Holder spaces 465
23.1.5 Sobolev spaces on manifolds with boundary 466
23.1.6 Uniform Cm regularity of dQ 467
23.1.7 Trace theorem 468
23.1.8 The general Sobolev type embedding theorems 469
23.1.9 Smoothness across interfaces 470
23.2 Regular elliptic boundary value problems 472
23.2.1 Regular elliptic boundary value problems in M.^_ 474
23.3 Boundary operators, adjoint problem and Green formula 475
23.3.1 Boundary operators in neighboring domains 477
23.3.2 The Green formula for the operator L = AP 479
23.4 Elliptic boundary value problems 479
23.4.1 A priori estimates in Sobolev spaces 480
23.4.2 Fredholm operator 480
23.4.3 Elliptic boundary value problems in Holder spaces 482
23.4.4 Schauder s continuous parameter method 483
23.5 Bibliographical notes 484
24 Afterword 485
Bibliography to Part IV 487
Index 491
|
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| id | DE-604.BV014119484 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:58:00Z |
| institution | BVB |
| isbn | 0124224903 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009674853 |
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| physical | XIV, 498 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
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| publisher | Acad. Press |
| record_format | marc |
| spelling | Kunčev, Ognjan I. Verfasser aut Multivariate polysplines applications to numerical and wavelet analysis Ognyan Kounchev San Diego [u.a.] Acad. Press 2001 XIV, 498 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Differential equations, Elliptic Numerical solutions Polyharmonic functions Spline theory Mehrere Variable (DE-588)4277015-4 gnd rswk-swf Spline (DE-588)4182391-6 gnd rswk-swf Spline (DE-588)4182391-6 s Mehrere Variable (DE-588)4277015-4 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009674853&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kunčev, Ognjan I. Multivariate polysplines applications to numerical and wavelet analysis Differential equations, Elliptic Numerical solutions Polyharmonic functions Spline theory Mehrere Variable (DE-588)4277015-4 gnd Spline (DE-588)4182391-6 gnd |
| subject_GND | (DE-588)4277015-4 (DE-588)4182391-6 |
| title | Multivariate polysplines applications to numerical and wavelet analysis |
| title_auth | Multivariate polysplines applications to numerical and wavelet analysis |
| title_exact_search | Multivariate polysplines applications to numerical and wavelet analysis |
| title_full | Multivariate polysplines applications to numerical and wavelet analysis Ognyan Kounchev |
| title_fullStr | Multivariate polysplines applications to numerical and wavelet analysis Ognyan Kounchev |
| title_full_unstemmed | Multivariate polysplines applications to numerical and wavelet analysis Ognyan Kounchev |
| title_short | Multivariate polysplines |
| title_sort | multivariate polysplines applications to numerical and wavelet analysis |
| title_sub | applications to numerical and wavelet analysis |
| topic | Differential equations, Elliptic Numerical solutions Polyharmonic functions Spline theory Mehrere Variable (DE-588)4277015-4 gnd Spline (DE-588)4182391-6 gnd |
| topic_facet | Differential equations, Elliptic Numerical solutions Polyharmonic functions Spline theory Mehrere Variable Spline |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009674853&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kuncevognjani multivariatepolysplinesapplicationstonumericalandwaveletanalysis |