Approximation Theory, Spline Functions and Applications

These are the Proceedings of the NATO Advanced Study Institute on Approximation Theory, Spline Functions and Applications held in the Hotel villa del Mare, Maratea, Italy between April 28,1991 and May 9, 1991. The principal aim of the Advanced Study Institute, as reflected in these Proceedings, was to bring together recent and up-to-date developments of the subject, and to give directions for future research. Amongst the main topics covered during this Advanced Study Institute is the subject of uni­ variate and multivariate wavelet decomposition over spline spaces. This is a relatively new area in approximation theory and an increasingly impor­ tant subject. The work involves key techniques in approximation theory­ cardinal splines, B-splines, Euler-Frobenius polynomials, spline spaces with non-uniform knot sequences. A number of scientific applications are also highlighted, most notably applications to signal processing and digital im­ age processing. Developments in the area of approximation of functions examined in the course of our discussions include approximation of periodic phenomena over irregular node distributions, scattered data interpolation, Pade approximants in one and several variables, approximation properties of weighted Chebyshev polynomials, minimax approximations, and the Strang­ Fix conditions and their relation to radial functions. I express my sincere thanks to the members of the Advisory Commit­ tee, Professors B. Beauzamy, E. W. Cheney, J. Meinguet, D. Roux, and G. M. Phillips. My sincere appreciation and thanks go to A. Carbone, E. DePas­ cale, R. Charron, and B.

Inhalt

Preface. Approximation by Functions of Nonclassical Form; E.W. Cheney. Wavelets- with Emphasis on Spline-Wavelets and Applications to Signal Analysis; C.K. Chui. Padé Approximation in One and More Variables; A. Cuyt. Rational Hermite Interpolation in One and More Variables; A. Cuyt. The Method of Alternating Orthogonal Projections; F. Deutsch. Selections for Metric Projections; F. Deutsch. Weighted Polynomials; M. v. Golitschek. Some Aspects of Radial Basis Function Approximation; W.A. Light. Using the Refinement Equation for the Construction of Pre-Wavelets VI: Shift Invariant Subspaces; C.A. Micchelli. A Tutorial on Multivariate Wavelet Decomposition; C.A. Micchelli. Error Estimates for Near-Minimax Approximations; G.M. Phillips. Different Metrics and Location Problems; E. Casini, P.L. Papini. On the Effectiveness of Some Inversion Methods for Noisy Fourier Series; L. De Michele, M. Di Natale, D. Roux. A Generalization of N-Widths; A.G. Aksoy. The Equivalence of the Usual and Quotient Topologies for CINFINITY(E) when E£sub£Rn is Whitney p-Regular; L.P. Bos, P.D. Milman. Korovkin Theorems for Vector-Valued Continuous Functions; M. Campiti. On Modified Bojanic-Shisha Operators; A.S. Cavaretta, S.S. Guo. A Property of Zeros and Cotes Numbers of Hermite and Laguerre Orthogonal Polynomials; F. Costabile. Hermite-Fejer and Hermite Interpolation; G. Criscuolo, B. Della Vecchia, G. Mastroianni. New Results on Lagrange Interpolation; G. Criscuolo, G. Mastroianni. Ambiguous Loci in Best Approximation Theory; F.S. De Blasi, J. Myjak. A Theorem on Best Approximations in Topological Vector Spaces; E. De Pascale, G. Trombetta. On theCharacterization of Totally Positive Matrices; M. Gasca, J.M. Pena. Iterative Methods for the General Order Complementarity Problem; G. Isac. Wavelets, Splines, and Divergence-Free Vector Functions; P-G. Lemarie-Rieusset. An Approach to Meromorphic Approximation in a Stein Manifold; C.H. Lutterodt. Approximating Fixed Points for Nonexpansive Maps in Hilbert Spaces; G. Marino. On Approximation and Interpolation of Convex Functions; M. Neamtu. Convergence of Approximating Fixed Point Sets for Multivalued Nonexpansive Mappings; P. Pietramala. A Subdivision Algorithm for Non-Uniform B-Splines; R. Qu, J.A. Gregory. Some Applications of an Approximation Theorem for Fixed Points of Multi-Valued Contractions; B. Ricceri. Geometrical Differentiation and High-Accuracy Curve Interpolation; R. Schaback. On Best Simultaneous Approximation in Normed Linear Spaces; V.M. Sehgal, S.P. Singh. Some Examples Concerning Projection Constants; B. Shekhtman. Subject Index.