Approximation Theory IX

Volume 2: Computational Aspects

Charles Chui and Larry Schumaker, eds

Vanderbilt University Press, Nashville

ISBN 0-8265-1326-3

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . ix

On the Reproduction of Linear Functions by Local Trigonometric Bases Kai Bittner . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Discretization of Nonlinear Terms Using Biorthogonal Wavelets: Hybrid Formulations

José E. Castilho and Sônia M. Gomes . . . . . . . . . . . . . . 9

Interpolation by Cubic Splines on Triangulations Oleg Davydov, Günther Nürnberger, and Frank Zeilfelder . . . . . 17

Interpolation and Almost Interpolation by Weak Chebyshev Spaces Oleg Davydov and Manfred Sommer . . . . . . . . . . . . . . 25

A Hermite Subdivision Scheme for the Evaluation of the Powell-Sabin 12-Split Element

Nira Dyn and Tom Lyche . . . . . . . . . . . . . . . . . . . 33

Fast Fourier Transforms for Nonequispaced Data Benedikte Elbel and Gabriele Steidl . . . . . . . . . . . . . . . 39

Scattered Data Interpolation by C^{2} Quintic Splines Using Energy Minimization

Kathleen W. Farmer and Ming-Jun Lai . . . . . . . . . . . . . 47

On Smoothing for Multilevel Approximation with Radial Basis Functions

Gregory E. Fasshauer . . . . . . . . . . . . . . . . . . . . 55

A Semi-Prewavelet Approach to Piecewise Linear Prewavelets on Triangulations

Michael S. Floater and Ewald G. Quak . . . . . . . . . . . . . 63

Refinable Spline Functions Tim N. T. Goodman . . . . . . . . . . . . . . . . . . . . . 71

How is a Vector Pyramid Scheme Affected by Perturbation in the Mask?

Bin Han and Thomas A. Hogan . . . . . . . . . . . . . . . . 97 Approximation to Small Deformation of Surfaces and Its Application

Matthew He and Chandra Kambhamettu . . . . . . . . . . . . 105

Biorthogonal Spline Approximation and Its Wavelet Analysis Tian Xiao He . . . . . . . . . . . . . . . . . . . . . . . . . 113

A New Sufficient Condition for the Orthonormality of Refinable Functions

Wenjie He and Ming-Jun Lai . . . . . . . . . . . . . . . . . 121

Optimal Triangulations and Smoothness Conditions for Bivariate Splines

Don Hong, Huan-Wen Liu, and Ram Mohapatra . . . . . . . . . 129

Norming Sets and Scattered Data Approximation on Spheres Kurt Jetter, Joachim Stöckler, and Joseph D. Ward . . . . . . . . 137

Radial Quasi Interpolation on S^{2} A. K. Kushpel and J. Levesley . . . . . . . . . . . . . . . . . 145

Bivariate Spline Method for Navier-Stokes Equations: Domain Decomposition Technique

Ming-Jun Lai and Paul Wenston . . . . . . . . . . . . . . . . 153

Divergence-Free Multiwavelets Joseph D. Lakey, Peter R. Massopust, and Maria C. Pereyra …. 161

On Intra- and Inter-orthogonal Scaling and Wavelet Vectors Jian-ao Lian . . . . . . . . . . . . . . . . . . . . . . . . . 169

General Moments Evaluation of Scaling Functions and Wavelets Jian-ao Lian and George A. Roberts . . . . . . . . . . . . . . . 179

Properly Posed Set of Nodes for Bivariate Lagrange Interpolation Xue-Zhang Liang and Chun-Mei Lu . . . . . . . . . . . . . . . 189

On the Closure of the Union of Nested Subspaces of L^{2}(IR^{d})

R. A. Lorentz and W. R. Madych . . . . . . . . . . . . . . . . 197

Calculating Joint Spectral Radius of Matrices and Hölder Exponent of Wavelets

Mohsen Maesumi . . . . . . . . . . . . . . . . . . . . . . 205

On Interpolation by Hermite Tension Splines of Arbitrary Order Miljenko Marušić . . . . . . . . . . . . . . . . . . . . . . . 213

Recent Developments in Approximation via Positive Definite Functions

Francis J. Narcowich . . . . . . . . . . . . . . . . . . . . . 221

Simulation-Based Modeling Alexa Nawotki and Hans Hagen . . . . . . . . . . . . . . . . 243

A Multiscale Method for the Evaluation of Wiener Integrals Erich Novak, Klaus Ritter, and Achim Steinbauer . . . . . . . . . 251 Spline Interpolation on Convex Quadrangulations

Günther Nürnberger and Frank Zeilfelder . . . . . . . . . . . . 259

Covariance Matrix Reduction Techniques for Neural Multi-Source Direction Finding

Teresa H. O’Donnell and Hugh L. Southall . . . . . . . . . . . 267

Multigrid Prolongations and Matrix Subdivision Peter Oswald . . . . . . . . . . . . . . . . . . . . . . . . . 275

Wavelets and Their Associated Operators Amos Ron . . . . . . . . . . . . . . . . . . . . . . . . . . 283

A Tool for Approximation in Bivariate Periodic Sobolev Spaces Frauke Sprengel . . . . . . . . . . . . . . . . . . . . . . . 319

Convergence of Biorthogonal Coifman Wavelet Systems Jun Tian and Raymond O. Wells, Jr. . . . . . . . . . . . . . . 327

Optimal Bivariate Crosscut Segmentations Guido Walz . . . . . . . . . . . . . . . . . . . . . . . . . 337

Variational Subdivision for Natural Cubic Splines Joe Warren and Henrik Weimer . . . . . . . . . . . . . . . . . 345

Fitting Conic Sections to Measured Data G. Alistair Watson . . . . . . . . . . . . . . . . . . . . . . 353

Numerical Solution of Variational Problems by Radial Basis Functions Holger Wendland . . . . . . . . . . . . . . . . . . . . . . . 361

Wavelet Denoising Using Generalized Cross Validation Norman Weyrich . . . . . . . . . . . . . . . . . . . . . . . 369

Mortar Mixed Finite Element Approximations for Elliptic and Parabolic Equations

Mary F. Wheeler and Ivan Yotov . . . . . . . . . . . . . . . 377