Approximation Theory IX

Volume 1: Theoretical Aspects Charles Chui and Larry Schumaker, eds Vanderbilt University Press, Nashville ISBN 0-8265-1325-5

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . ix The Degree of Convergence of Sequences of Multivariate Wavelet Type Operators

George A. Anastassiou . . . . . . . . . . . . . . . . . . . . . 1 Best m-Term Approximation of Functions from Classes MW_{q,α α α α α α α α α}^{r}

Alexander V. Andrianov and Vladimir N. Temlyakov . . . . . . . . 7 Necessary Conditions for Abadie’s Constraint Qualification and Best Approximation of Vector-Valued Functions

Martin Bartelt . . . . . . . . . . . . . . . . . . . . . . . 15 n-Widths of Multiplier Operators on Two-Point Homogeneous Spaces

B. Bordin, A. K. Kushpel, J. Levesley, and S. A. Tozoni . . . . . . 23 Some Old Problems on Polynomials with Integer Coefficients

Peter Borwein . . . . . . . . . . . . . . . . . . . . . . . . 31 On the Irrationality of ∑_{i=0}^{∞} q^{−i} ∏_{j=0}^{i}(1 + q^{−j}r + q^{−2j}s)

Peter B. Borwein and Ping Zhou . . . . . . . . . . . . . . . . 51 On a Weighted Recursive Rule

Michele Campiti . . . . . . . . . . . . . . . . . . . . . . . 59 Approximation of the Inverse Frame Operator

Peter Casazza and Ole Christensen . . . . . . . . . . . . . . . 67 The Bernstein Operator is the Closest Positive Operator to a Projection

Bruce L. Chalmers, Dany Leviatan, and Michael P. Prophet …. 75 Finite-Dimensional Action Constants

B. L. Chalmers and K. Pan . . . . . . . . . . . . . . . . . . 83 Approximation by Sums of Exponentials to Decay Functions Using Piecewise Linear Models

Maurice G. Cox, Helen E. Joyce, and John C. Mason . . . . . . . 89 A Differential Geometric Approach to Equidistributed Knots on Riemannian Manifolds

Uwe Depczynski and Joachim Stöckler . . . . . . . . . . . . . 97 The Role of the Strong Conical Hull Intersection Property in Convex Optimization and Approximation

Frank Deutsch . . . . . . . . . . . . . . . . . . . . . . . 105

Fractional Error Constants for Quadrature Formulas Kai Diethelm . . . . . . . . . . . . . . . . . . . . . . . . . 113

Generalized Bernstein-Erdős Conjecture P. Dragnev, D. Legg, and D. Townsend . . . . . . . . . . . . . 119

Non-convexity of the Set of Pólya Limits A. Egger, T. Lay, and G. D. Taylor . . . . . . . . . . . . . . . 127

Estimates for Some Positive Operators via Step-Weight Functions with Concave Squares

Michael Felten . . . . . . . . . . . . . . . . . . . . . . . 133

Polynomial Approximation on the m-Dimensional Ball Michael I. Ganzburg . . . . . . . . . . . . . . . . . . . . . . 141

Approximation Using Scale Transformations Christian Gout . . . . . . . . . . . . . . . . . . . . . . . . 149

Linear Discrete Operators and Recovery of Functions Ivan Ivanov and Boris Shekhtman . . . . . . . . . . . . . . . . 157

Chebyshev Quadrature Recognizes Algebraic Curves and Surfaces J. Korevaar . . . . . . . . . . . . . . . . . . . . . . . . . 165

Ray Sequences of Best Rational Approximants to Entire Functions A. V. Krot, V. A. Prokhorov, and E. B. Saff . . . . . . . . . . . 175

Extremal Problems for Logarithmic Potentials A. B. J. Kuijlaars . . . . . . . . . . . . . . . . . . . . . . . 201

Monotone Approximation Estimates Involving the Third Modulus of Smoothness

Dany Leviatan and Igor A. Shevchuk . . . . . . . . . . . . . . 223

Infinite Dimensional Width of Function Class With Non-symmetric Smoothness Conditions

Yongping Liu . . . . . . . . . . . . . . . . . . . . . . . . 231

Rational Approximation in Logarithmic Capacity of Meromorphic Maps from C^{n} to C^{m}

Clement Lutterodt and Stanley Einstein-Matthews . . . . . . . . 239

Best Approximation by Quadratic Algebraic Functions Allan W. McInnes . . . . . . . . . . . . . . . . . . . . . . 247

L_{p}-Convergence of the Lagrange-Type Interpolation in the Rational Space

G. Min . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Approximation of Functions by Linear Matrix Operators

M. L. Mittal and Neeraj Bhardwaj . . . . . . . . . . . . . . . 263

Weighted Approximation on Compact Sets Igor E. Pritsker . . . . . . . . . . . . . . . . . . . . . . . . 271

Characterization of Best L_{1}-Coapproximation Geetha S. Rao and R. Saravanan . . . . . . . . . . . . . . . . 279

Algebraic Aspects of Polynomial Interpolation in Several Variables Thomas Sauer . . . . . . . . . . . . . . . . . . . . . . . . 287

Adaptive Approximation and Compression Bl. Sendov . . . . . . . . . . . . . . . . . . . . . . . . . . 295

On the Discrete Norms of Polynomials Boris Shekhtman . . . . . . . . . . . . . . . . . . . . . . . 303

Generation of Weight Functions for Orthogonal Polynomials M.-R. Skrzipek . . . . . . . . . . . . . . . . . . . . . . . . 309

On Products of Positive Definite Functions Hans Strauss . . . . . . . . . . . . . . . . . . . . . . . . . 317

On the Jackson Theorem for Approximation by Algebraic Polynomials

Gancho Tachev . . . . . . . . . . . . . . . . . . . . . . . 325

On Approximation of Cauchy-Type Integrals by Sequences of Rational Functions with Preassigned Poles

Genrikh Ts. Tumarkin . . . . . . . . . . . . . . . . . . . . 331

Sharp Error Estimates for Multivariate Positive Linear Operators Shayne Waldron . . . . . . . . . . . . . . . . . . . . . . . 339

Realizable Approximation Bounds for a Solvable Neural Network Sumio Watanabe . . . . . . . . . . . . . . . . . . . . . . . 347