Atteia Hilbertian Kernels and Spline Functions

1. Auflage 2014
ISBN: 978-1-4832-9519-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 4, 393 Seiten, Web PDF

Reihe: Studies in Computational Mathematics

In this monograph, which is an extensive study of Hilbertian approximation, the emphasis is placed on spline functions theory. The origin of the book was an effort to show that spline theory parallels Hilbertian Kernel theory, not only for splines derived from minimization of a quadratic functional but more generally for splines considered as piecewise functions type.Being as far as possible self-contained, the book may be used as a reference, with information about developments in linear approximation, convex optimization, mechanics and partial differential equations.

Autoren/Hrsg.

Atteia, M.

Inhaltsverzeichnis

1;Front Cover;1
2;Hilbertian Kernels and Spline Functions;4
3;Copyright Page;5
4;Table of Contents;8
5;Acknowledgements;6
6;Introduction;12
7;Chapter I. HILBERTIAN KERNELS;14
7.1;1 - HILBERTIAN SUBSPACES OF RO AND ASSOCIATED KERNELS;17
7.2;2 - BIJECTION OF THE SET OF HILBERTIAN SUBSPACES OF RO ONTO R+O×O;24
7.3;3 - SUPPORT OF A KERNEL;31
7.4;4 - SCHWARTZ'S KERNEL OF A HILBERTIAN SUBSPACE OF RO (resp. of a LOCALLY CONVEX SPACE (l.c.s.).;34
7.5;5 - ORDERED CONVEX CONE STRUCTURE OF HILB (RO) (resp. HILB (E) );45
7.6;6 - IMAGE OF AN HILBERT SUBSPACE BY A LINEAR MAPPING AND ITS ASSOCIATED KERNEL;53
7.7;7 - ELEMENTARY OPERATIONS ONTO KERNELS OF HILBERTIAN SUBSPACES;57
7.8;8 - TENSOR PRODUCT OF HILBERTIAN SUBSPACES AND ASSOCIATED KERNELS;72
7.9;9 - QUOTIENT HILBERT SUBSPACES. ASSOCIATED SEMI-KERNELS.;87
7.10;10 - KERNELS REGULARITY;90
7.11;11 - HILBERTIAN SUBSPACES KERNELS AND QUADRATIC CONVEX FUNCTIONALS;93
7.12;12 - COMPLEX KERNELS. BOCHNER THEOREM;99
8;Chapter II. INTERPOLATION. APPROXIMATION OF LINEAR FUNCTIONALS;108
8.1;1 - A GENERAL FORMULATION OF THE INTERPOLATION PROBLEM J;109
8.2;2 - SARD'S FACTORIZATION THEOREM;111
8.3;3 - LAGRANGE INTERPOLATION;113
8.4;4 - BASIS CHANGE IN THE LAGRANGE INTERPOLATION;116
8.5;5 - NEWTON INTERPOLATION;119
8.6;6 - DUAL PROBLEM OF THE LAGRANGE INTERPOLATION. ERROR IN THE LAGRANGE INTERPOLATION;123
8.7;7 - EXAMPLES;125
8.8;8 - SOME PRELIMINARY RESULTS ABOUT THE BASES OF TOPOLOGICAL VECTOR SPACES;134
8.9;9 - INTERPOLATION WITH AN INFINITY OF DATA;143
8.10;10 - FRAMES;149
9;Chapter III. INTERPOLATING AND SMOOTHING SPLINE (OR SCHOENBERG) FUNCTIONS;158
9.1;1 - INTERPOLATING SPLINE FUNCTIONS;159
9.2;2. SMOOTHING SPLINE FUNCTIONS;165
9.3;3. SPLINE FUNCTIONS AND (SEMI-) HILBERTIAN KERNELS;169
9.4;4. EXAMPLES OF SPLINE FUNCTIONS (OR SCHOENBERG FUNCTIONS);177
10;Chapter IV. OPERATIONS ON SPLINE FUNCTIONS;188
10.1;1- IMAGE SPLINE FUNCTION;188
10.2;2- DIRECT HILBERT SUM AND CARTESIAN PRODUCT OF SPLINE FUNCTIONS;191
10.3;3 - SUM OF "SPLINE" FUNCTIONS;195
10.4;4 - TENSOR PRODUCT OF SPLINE FUNCTIONS;196
10.5;5 - RELATIONS BETWEEN THE OPERATIONS ON SPLINE FUNCTIONS AND OPERATIONS ON HILBERTIAN KERNELS;203
11;Chapter V. INTERNAL AND EXTERNAL CONVERGENCE OF SPLINE FUNCTIONS;210
11.1;1 - ABOUT INTERNAL AND EXTERNAL CONVERGENCES;210
11.2;2 - INTERNAL CONVERGENCE OF THE I.S.F.;212
11.3;3 - EXTERNAL CONVERGENCE OF THE I.S.F.;225
11.4;4 - CONVERGENCE OF TENSOR PRODUCT OF I.S.F.;229
11.5;5 - INTERNAL CONVERGENCE OF THE SMOOTHING SPLINE FUNCTIONS;235
12;Chapter VI. SPLINE FUNCTIONS ONTO A CONVEX SET;244
12.1;1 - INTERPOLATING SPLINE FUNCTION ON A CONVEX SET. PRIMAL PROBLEM. FIRST PROPERTIES;245
12.2;2. CONDITIONS FOR CLOSURE OF u(G);247
12.3;3. STUDY OF THE SET OG;254
12.4;4 - EQUIVALENCE OF THE PROBLEM g AND A CONVEX OPTIMIZATION PROBLEM WITHOUT CONSTRAINT;263
12.5;5 - DUALITY;266
12.6;6. SMOOTH SPLINE FUNCTION ON A CONVEX SET;281
13;Chapter VII. SPLINE MANIFOLDS AND LINEAR ELASTICITY;292
13.1;1 - ELEMENTARY NOTIONS OF TRIMENSIONAL ELASTICITY;294
13.2;2 - LINEAR ELASTICITY AND SPLINE FUNCTIONS;296
13.3;3 - SPLINE MANIFOLD ON AN OPEN SET O;301
13.4;4 - SPLINE MANIFOLD RELATED TO PARTIAL DIFFERENTIAL OPERATOR;310
14;Chapter VIII. B-SPLINES, BOX-SPLINES, SIMPLICIAL SPLINES;340
14.1;1 - THE HILBERT SPACE Lm;341
14.2;2 - B-SPLINE;342
14.3;3 - BOX SPLINES - BASIC PROPERTIES;355
14.4;4 - EXAMPLES OF BOX-SPLINES;358
14.5;5 - THE BOX-SPLINES REVISITED;362
14.6;6 - GENERALIZZED BARYCENTRIC COORDINATES;370
14.7;7 - HILBERTIAN REPRESENTATION OF A SIMPLICIAL COMPLEX;371
14.8;8 - A BASIC APPLICATION;374
14.9;9 - ELEMENTARY OPERATIONS ON HILBERTIAN REPRESENTATIONS OF SIMPLICIAL COMPLEXES;376
15;SOME COMMENTS;382
16;BIBLIOGRAPHY;386