Approximation Theory and Spline Functions

A NATO Advanced Study Institute on Approximation Theory and Spline Functions was held at Memorial University of Newfoundland during August 22-September 2, 1983. This volume consists of the Proceedings of that Institute. These Proceedings include the main invited talks and contributed papers given during the Institute. The aim of these lectures was to bring together Mathematicians, Physicists and Engineers working in the field. The lectures covered a wide range including ~1ultivariate Approximation, Spline Functions, Rational Approximation, Applications of Elliptic Integrals and Functions in the Theory of Approximation, and Pade Approximation. We express our sincere thanks to Professors E. W. Cheney, J. Meinguet, J. M. Phillips and H. Werner, members of the International Advisory Committee. We also extend our thanks to the main speakers and the invi ted speakers, whose contri­ butions made these Proceedings complete. The Advanced Study Institute was financed by the NATO Scientific Affairs Division. We express our thanks for the generous support. We wish to thank members of the Department of Mathematics and Statistics at MeMorial University who willingly helped with the planning and organizing of the Institute. Special thanks go to Mrs. Mary Pike who helped immensely in the planning and organizing of the Institute, and to Miss Rosalind Genge for her careful and excellent typing of the manuscript of these Proceedings.

Table of Contents:
Products of Polynomials.- Exchange Algorithms, Error Estimations and Strong Unicity in Convex Programming and Chebyshev Approximation.- Four Lectures on Multivariate Approximation.- The Approximation of Certain Functions by Compound Means.- A Practical Method for Obtaining a Priori Error Bounds in Pointwise and Mean-Square Approximation Problems.- Surface Spline Interpolation Basic Theory and Computational Aspects.- Interpolation of Scattered Data Distance Matrices and Conditionally Positive Definite Functions.- Semi-Norms in Polynomial Approximation.- On Spaces of Piecewise Polynomials in Two Variables.- Birkhoff Interpolation on the Roots of Unity.- Applications of Transformation Theory A Legacy from Zolotarev (1847–1878).- Explicit Algebraic Nth Order Approximations to PI.- Solving Integral Equations of Nuclear Scattering by Splines.- H-Sets for Non-Linear Constrained Approximation.- Operator Padé Approximants Some Ideas Behind the Theory and a Numerical Illustration.- Harmonic Approximation.- Best Harmonic L1 Approximation to Subharmonic Functions.- B-Splines on the Circle And Trigonometric B-Splines.- On Reducing the Computational Error in the Successive Approximations Method.- Lebesgue Constants Determined by Extremal Sets.- Error Bounds for Interpolation by Fourth Order Trigonometric Splines.- Approximation of Derivatives in Rn Application Construction of Surfaces in R2.- Meromorphic Functions, Maps and Their Rational Approximantsin Cn..- Splines and Collocation for Ordinary Initial Value Problems.- Degree of Approximation of Quasi-Hermite-Fejer Interpolation Based on Jacobi Abscissas Pn(?,?) (x).- Using Inclusion Theorems to Establish the Summability of Orthogonal Series.- On Projections in Approximation Theory.- A Survey of Exterior Asymptotics forOrthogonal Polynomials Associated With a Finite Interval and a Study of the Case of the General Weight Measures.- List of Participants.