Numerical Methods in Approximation Theory, Vol. 9

1. Front Matter

   Pages i-xiv

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2. Blending Approximations with Sine Functions

   • G. Baszenski, F.-J. Delvos, S. Jester

   Pages 1-19

3. Quasi-interpolation in the Absence of Polynomial Reproduction

   • R. K. Beatson, W. A. Light

   Pages 21-39

4. Estimating the Condition Number for Multivariate Interpolation Problems

   • P. Binev, K. Jetter

   Pages 41-52

5. Wavelets on a Bounded Interval

   • Charles K. Chui, Ewald Quak

   Pages 53-75

6. Quasi-Kernel Polynomials and Convergence Results for Quasi-Minimal Residual Iterations

   • Roland W. Freund

   Pages 77-95

7. Rate of Approximation of Weighted Derivatives by Linear Combinations of SMD Operators

   • Margareta Heilmann

   Pages 97-115

8. Approximation by Multivariate Splines: an Application of Boolean Methods

   • Rong-Qing Jia

   Pages 117-134

9. L^m,ℓ,s-splines in ℝ^d

   • A. Le Méhauté, A. Bouhamidi

   Pages 135-154

10. Constructive Multivariate Approximation with Sigmoidal Functions and Applications to Neural Networks

    • Burkhard Lenze

    Pages 155-175

11. Spline-Wavelets of Minimal Support

    • T. Lyche, K. Mørken

    Pages 177-194

12. Necessary Conditions for Local Best Chebyshev Approximations by Splines with Free Knots

    • Bernd Mulansky

    Pages 195-206

13. C1 Interpolation on Higher-Dimensional Analogues of the 4-Direction Mesh

    • Andy Neff, Jörg Peters

    Pages 207-220

14. Tabulation of Thin Plate Splines on a Very Fine Two-Dimensional Grid

    • M. J. D. Powell

    Pages 221-244

15. The L2-Approximation Orders of Principal Shift-Invariant Spaces Generated by a Radial Basis Function

    • Amos Ron

    Pages 245-268

16. A Multi-Parameter Method for Nonlinear Least-Squares Approximation

    • R. Schaback

    Pages 269-283

17. Analog VLSI Networks

    • Walter Schempp

    Pages 285-300

18. Converse Theorems for Approximation on Discrete Sets II

    • G. Schmeisser

    Pages 301-316

19. A Dual Method for Smoothing Histograms Using Nonnegative C1-Splines

    • Jochen W. Schmidt

    Pages 317-329

20. Segment Approximation Using Linear Functionals

    • Manfred Sommer

    Pages 331-346

This book is the official proceedings of a conference on Numerical Methods in Approximation Theory which was held at the Mathematisches Forschungs­ institut in Oberwolfach during the week of November 24~30, 1991. It contains refereed and edited papers by 20 of the 49 participants. The book is dedicated to the memory of Prof. Lothar Collatz who main­ tained a long and active interest in numerical approximation. It is the ninth in a series of volumes published by Birkhiiuser resulting from conferences on the subject held at Oberwolfach, and co-organized by Prof. Collatz. We now briefly describe the contents of the book. The paper of BASZEN­ SKI, DELVOS and JESTER deals with blending using sine double series expan­ sions of functions defined on the unit square. In addition to giving explicit error estimates for partial sums and for interpolating sine polynomials, they also show that Boolean sums yield almost the same asymptotic error estimates as the conventional tensor-product approach, but with a reduced number of terms. The paper of BEATSON and LIGHT discusses approximation by quasi­ interpolants which are sums of scaled translates of a one-parameter family of functions. They do not require reproduction of low degree polynomials, but nevertheless are able to give error bounds and analyze quasi-interpolation based on Gaussians and exponentials. BINEV and JETTER deal with multivariate interpolation using shifts of a single basis function. They treat both gridded data and scattered data. As examples, they consider box splines and certain radial basis functions.

• Invariant
• addition
• approximation
• derivative
• function
• numerical method
• theorem

• Fakultät und Institut für Mathematik, Ruhr-Universität Bochum, Bochum 1, Germany

  Dietrich Braess

• Dept. of Mathematics, Vanderbilt University, Nashville, USA

  Larry L. Schumaker

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