Trends in Approximation Theory

Kirill Kopotun, Tom Lyche, and Marian Neamtu, eds.

Vanderbilt Press, Nashville, 2001

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

Approximation Algebras and Applications J. M. Almira and U. Luther . . . . . . . . . . . . . . . . . . . 1

Nonpositive Jackson-type Approximations to Definite Integrals G. G. Anastassiou and S. G. Gal . . . . . . . . . . . . . . . . 11

Interpolatory Biorthogonal Multiwavelet Transforms Based on Hermite Splines

A. Averbuch and V. Zheludev . . . . . . . . . . . . . . . . . 19

Hausdorff Strong Unicity in Vector-Valued Chebyshev Approximation on Finite Sets

M. Bartelt . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Approximate Characteristics of Multiplier Operators on the Sphere

B. Bordin, A. K. Kushpel, and S. A. Tozoni . . . . . . . . . . . 39

On Minimal, Almost Locally Minimal, and Orthogonal Minimal Projections

B. L. Chalmers and B. Shekhtman . . . . . . . . . . . . . . . 49

A Look at Bases in Function Spaces at the Beginning of the XXI Century

Z. Ciesielski . . . . . . . . . . . . . . . . . . . . . . . . . 53

Property A and Uniqueness in Best Approximation by n-Convex Functions

A. Damas and M. Marano . . . . . . . . . . . . . . . . . . . 73

On Computation of Stable Local Bases for Bivariate Polynomial Splines

O. Davydov . . . . . . . . . . . . . . . . . . . . . . . . . 83

Spline Subdivision Schemes for Compact Sets with Metric Averages N. Dyn and E. Farkhi . . . . . . . . . . . . . . . . . . . . . 93

Nonsymmetric Multilevel RBF Collocation within an Operator Newton Framework for Nonlinear PDEs

G. E. Fasshauer . . . . . . . . . . . . . . . . . . . . . . . 103

New Limit Relations for Polynomial Approximations M. I. Ganzburg . . . . . . . . . . . . . . . . . . . . . . . 113 The L_{∞}-norm of Orthogonal and Interpolation Projections

M. v. Golitschek . . . . . . . . . . . . . . . . . . . . . . . 123

Partially Greedy Algorithms R. Gribonval . . . . . . . . . . . . . . . . . . . . . . . . . 143

Designing a G^{1} A-spline Surface over a Triangulation V. Hernándes Mederos and J. Estrada Sarlabous . . . . . . . . . 149

On an Algorithm of Remez Type K. Kitahara and Y. Sakamori . . . . . . . . . . . . . . . . . 161

Tauberian Theory, Approximation and Lacunary Series of Powers J. Korevaar . . . . . . . . . . . . . . . . . . . . . . . . . 169

Symmetry Properties of Cardinal Interpolation L-splines and Polysplines

O. Kounchev and H. Render . . . . . . . . . . . . . . . . . . 191

n-Widths of Sobolev’s Classes on Compact Globally Symmetric Spaces of Rank One

A. Kushpel . . . . . . . . . . . . . . . . . . . . . . . . . 203

The Nonexistence of Bivariate Symmetric Wavelets with Short Support and Two Vanishing Moments

M.-J. Lai and D. W. Roach . . . . . . . . . . . . . . . . . . 213

Trivariate C^{1} Cubic Splines for Numerical Solution of Biharmonic Equations

M.-J. Lai and P. Wenston . . . . . . . . . . . . . . . . . . . 225

Representation of Natural Terrain by Cubic L_{1} Splines J. E. Lavery and D. E. Gilsinn . . . . . . . . . . . . . . . . . 235

Polynomial Identities of Bezout Type J.-ao Lian . . . . . . . . . . . . . . . . . . . . . . . . . . 243

On Local Wavelet Coefficients of Functions of Generalized Lipschitz Classes

J. Lippus . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Bivariate Cardinal Spline Functions for Digital Signal Processing X. Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Values and Ethics in Mathematics G. G. Lorentz . . . . . . . . . . . . . . . . . . . . . . . . 273

Diagonal Pade Sequences for Functions Meromorphic in the Unit Ball Approximate Well near Zero

D. S. Lubinsky . . . . . . . . . . . . . . . . . . . . . . . . 297

Irregularly Sampled Band Limited Functions, Riesz Bases, and Piecewise Polynomial Splines

W. R. Madych . . . . . . . . . . . . . . . . . . . . . . . . 307 Improved Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels

T. M. Morton . . . . . . . . . . . . . . . . . . . . . . . . 317

Boundary Value Problems with Algebraic Polynomials S. M. Nikol’skii . . . . . . . . . . . . . . . . . . . . . . . 327

Local Lagrange Interpolation by Cubic Splines on a Class of Triangulations

G. Nürnberger and F. Zeilfelder . . . . . . . . . . . . . . . . 333

Extremal Properties of Trigonometric Polynomials with Applications to Signal Design

P. Oswald . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Frames of Translates in Abstract Hilbert Spaces and the Generalized Frame Multiresolution Analysis

M. Papadakis . . . . . . . . . . . . . . . . . . . . . . . . 353

Linear Transport Equations with Continuous Solutions B. Popov . . . . . . . . . . . . . . . . . . . . . . . . . . 363

On Rational Approximation of Meromorphic Functions V. A. Prokhorov . . . . . . . . . . . . . . . . . . . . . . . 373

Approximation of Continuous Functions by Convolution J. B. Prolla and M. S. Kashimoto . . . . . . . . . . . . . . . 383

Construction of Nonnegative Volume Matching Splines J. W. Schmidt, M. Bastian, and B. Mulansky . . . . . . . . . . 391

Approximation of Functions by Means of the Poweroid Operators S_{m,r,s}^{α}

D. D. Stancu and J. W. Drane . . . . . . . . . . . . . . . . . 401

Approximation by Conditionally Positive Definite Functions H. Strauss . . . . . . . . . . . . . . . . . . . . . . . . . . 407

Gaussian Interpolation Revisited H. Wendland . . . . . . . . . . . . . . . . . . . . . . . . 417

Approximation by Conditionally Positive Definite Functions with Finitely Many Centers

J. Yoon . . . . . . . . . . . . . . . . . . . . . . . . . . . 427