New Developments in Approximation Theory: 2nd International Dortmund Meeting (IDoMAT) ’98, Germany, February 23–27, 1998 PDF
Preview New Developments in Approximation Theory: 2nd International Dortmund Meeting (IDoMAT) ’98, Germany, February 23–27, 1998
ISNM International Series of Numerical Mathematics Vol. 132 Managing Editors: K.-H. Hoffmann, München D. Mittelmann, Tempe Associate Editors: R. E. Bank, La Jolla H. Kawarada, Chiba R. J. LeVeque, Seattle C. Verdi, Milano Honorary Editor: J. Todd, Pasadena New Developments in Approximation Theory 2nd International Dortmund Meeting (IDoMAT) '98, Germany, February 23-27,1998 Edited by M.W. Müller M.D. Buhmann D.H. Mache M. Feiten Springer Basel AG Editors: current address of Detlef H. Mache: Manfred W. Müller Ludwig-Maximilians-Universität München Martin D. Buhmann Numerische Analysis Detlef H. Mache Theresienstrasse 39 Michael Feiten 80333 München Lehrstuhl VIII für Mathematik Germany Universität Dortmund Vogelpothsweg 87 44221 Dortmund Germany 1991 Mathematics Subject Classification 65Dxx A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bibliothek Cataloging-in-Publication Data New developments in approximation theory / 2nd International Dortmund Meeting (IDoMAT) '98, Germany, February 23-27, 1998 / ed. by M. W. Müller ... - Springer Basel AG, 1999 (International series of numerical mathematics ; Vol. 132) ISBN 978-3-0348-9733-4 ISBN 978-3-0348-8696-3 (eBook) DOI 10.1007/978-3-0348-8696-3 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 1999 Springer Basel AG Originally published by Birkhäuser Verlag in 1999 Softcover reprint of the hardcover 1st edition 1999 Printed on acid-free paper produced of chlorine-free pulp. TCF oo Cover design: Heinz Hiltbrunner, Basel ISBN 978-3-0348-9733-4 98765432 1 Preface This book contains the refereed papers which were presented at the second In ternational Dortmund Meeting on Approximation Theory (IDoMAT 98) at Haus Bommerholz, the conference center of Dortmund University, during the week of February 23-27,1998. At this conference 52 researchers and specialists from Bul garia, China, France, Great Britain, Hungary, Israel, Italy, Roumania, South Africa and Germany participated and described new developments in the fields of uni variate and multivariate approximation theory. The papers cover topics such as radial basis functions, bivariate spline interpolation, multilevel interpolation, mul tivariate triangular Bernstein bases, Pade approximation, comonotone polynomial approximation, weighted and unweighted polynomial approximation, adaptive ap proximation, approximation operators of binomial type, quasi interpolants, gen eralized convexity and Peano kernel techniques. This research has applications in areas such as computer aided geometric design, as applied in engineering and medical technology (e.g. computerised tomography). Again this international conference was wholly organized by the Dortmund Lehrstuhl VIII for Approximation Theory. The organizers attached great impor tance to inviting not only well-known researchers but also young talented math ematicians. IDoMAT 98 gave an excellent opportunity for talks and discussions between researchers from different fields of Approximation Theory. In this way the conference was characterized by a warm and cordial atmosphere. The success of IDoMAT 98 was above all due to everyone of the participants. Our thanks go to the referees for their prompt cooperation as well as their accurate work, so that we can present in this volume an interesting impression of the good quality of our meeting. Finally we would like to thank Deutsche Forschungsgemeinschaft for the fi nancial support and Birkhauser-Verlag for agreeing to publish the proceedings in the ISNM series. The editors March 1999 Contents Feller Semigroups, Bernstein type Operators and Generalized Convexity Associated with Positive Projections Francesco Altomare .................................................. 9 Gregory's Rational Cubic Splines in Interpolation Subject to Derivative Obstacles Marion Bastian-Walther and Jochen W. Schmidt 33 Interpolation by Splines on Triangulations Oleg Davydov, Gunther Nurnberger and Prank ZeilJelder 49 On the Use of Quasi-Newton Methods in DAB-Codes Christoph Flredebeul, Christoph Weber . . . . . . .. . . . . . . .. . . . . . .. . .. . . . .. . 71 On the Regularity of Some Differential Operators Karsten Kamber and Xinlong Zhou .................................. 79 Some Inequalities for Trigonometric Polynomials and their Derivatives Hans-Bernd Knoop and Xinlong Zhou ............................... 87 Inf-Convolution and Radial Basis Functions Alain Le M ehaute ................................................... 95 On a Special Property of the Averaged Modulus for Functions of Bounded Variation Burkhard Lenze ...................................................... 109 A Simple Approach to the Variational Theory for Interpolation on Spheres Jeremy Levesley, Will Light, David Ragozin and Xingping Sun.. . .. . . 117 Constants in Comonotone Polynomial Approximation - A Survey L. Leviatan and I.A. Shevchuk ....................................... 145 Will Ramanujan kill Baker-Gammel-Wills? (A Selective Survey of Pade Approximation) Doran S. Lubinsky ................................................. .. 159 Approximation Operators of Binomial Type Alexandru Lupa§ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 175 8 Contents Certain Results involving Gammaoperators Alexandru Lupa§, Detlef-H. Mache, Volker Maier, Manfred-W. Muller.................................................. 199 Recent research at Cambridge on radial basis functions M. J. D. Powell ..................................................... 215 Representation of quasi-interpolants as differential operators and applications Paul Sablonniere .................................................... 233 Native Hilbert Spaces for Radial Basis Functions I Robert Schaback ..................................................... 255 Adaptive Approximation with Walsh-similar Functions Bl. Sendov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 283 Dual Recurrence and Christoffel-Darboux-Type Formulas for Orthogonal Polynomials Michael - Ralf Skrzipek .............................................. 303 On Some Problems of Weighted Polynomial Approximation and Interpolation J6zsef Szabados ...................................................... 315 Asymptotics of derivatives of orthogonal polynomials based on generalized Jacobi weights. Some new theorems and applications Peter Vertesi ........................................................ 329 List of participants ....................................................... 341 International Series of Numerical Mathematics Vol. 132, © 1999 Birkhiiuser Verlag Basel/Switzerland Feller Semigroups, Bernstein type Operators and Generalized Convexity Associated with Positive Projections Dedicated to Professor Giuseppe Mastroianni on the occasion of his 60th birthday Francesco Altomare* - loan Rasa Abstract We study the majorizing approximation properties of both Bernstein type operators and the corresponding Feller semigroups associated with a positive projection acting on the space of all continuous functions defined on a convex compact set. The relationship between these properties and some generalized form of convexity is investigated as well. 1 Introduction Starting from a positive linear projection acting on the space of all continuous func tions defined on a convex compact subset of a locally convex space, it is possible to construct several positive linear approximation methods such as Lototsky-Schnabl operators and Bernstein-Schnabl operators. Under suitable assumptions, these operators determine a Feller semigroup whose generator (A, D(A)) can be described in a core of its domain. Indeed, in the finite dimensional case, (A, D(A)) is the closure of a degenerate second-order elliptic differential operator. A detailed analysis of these operators, of the associated Feller semigroups and of their corresponding Markov processes can be found in the monograph [4]. During the last years, for both theoretical and practical questions, some at tention was also devoted to investigating the majorizing approximation properties 'The contribution of the first author is due to work done under the auspices of the G.N.A.F.A (C.N.R) and partially supported by Ministero dell' Universita R.S.T. (Quote 60% and 40%). The work was carried out in Oct. 1997 while the second author was Visiting Professor at the University of Bari. 9 10 Francesco Altomare - loan Rasa of Bernstein-Lototsky-Schnabl operators and of their corresponding Feller semi group, ([15], [16]); see also [2]). These properties have interesting applications in the study of the qualitative properties of the solutions of the initial-boundary problems associated with the operator (A, D(A)) and of the corresponding probability transition function. Moreover, it turns out that these properties are strongly related to a gener alized form of convexity, whose study seems to be of independent interest. In this paper we survey some known results on this topic and in addition, we present some new advances together with some open problems. 2 Differential operators associated with positive projections Let K be a convex compact subset of RP, p ;::: 1, having non-empty interior. We shall denote by C(K) the Banach lattice of all real-valued continuous functions on K endowed with the sup-norm and the natural order. The symbol C2(K) stands for the subspace of all functions f E C(K) which are two times continuously differentiable in the interior int K of K and whose partial derivatives of order ~ 2 can be continuously extended to K. For every u E C2(K) and i,j = 1, ... ,p we shall continue to denote by g~ and a~i2a~j the continuous extensions to K of the partial derivatives g~ and a~i2a~j defined on int K. For every j = 1, ... ,p we shall denote by prj E C (K) the continuous function on K defined by for each x = (xihs;iS;p E K. (1) As a starting point for the construction of the differential operators in question we fix a positive linear projection T: C(K) -+ C(K), i.e., T is a positive linear operator such that ToT = T. We also assume that T is not trivial, i.e., T is not the identity operator on C(K), and moreover, that T(l) = 1 (2) and (j = 1, ... ,p), (3) (here 1 denotes the function having constant value 1). Furthermore, set HT := T(C(K)) = {h E C(K) I Th = h} (4) and assume that for every z E K, a E [0,1] and h E HT we have (5) Feller Semigroups, Bernstein type Operators and Generalized Convexity 11 where hz,a/x) := h(ax + (1-a)z) (x E K). (6) If we denote by oeK and OHTK the set of extreme points of K and the Choquet boundary of K with respect to HT (see, e.g., [4, sect. 2.6]) we have (7) (see [4, sect. 6.1]). Moreover, we know ([4, Prop. 3.3.1 and Prop. 3.3.2]) that OHTK = {x E K I T(J)(x) = f(x) for every f E C(K)} (8) = {x E K I T(e)(x) = e(x)}, where p e(x) := IIxl12 = I>~ (9) i=l In particular, from (8) it follows that for every f, g E C(K) we have T(J) = T(g) (10) We are now in the position to introduce the differential operator WT C2(K) -t C(K) which is defined by (11) for every U E C2(K) and x E K, where for each i,j = 1, ... ,p (12) The operator WT is elliptic and degenerates on OHTK (in particular, on OeK), i.e. p L aij(x) ~i~j ~ 0 (13) i,j=l and WT(U)(x) = 0 (14) The operator WT will be called the elliptic second order differential operator associated with the projection T. Here we indicate some (fundamental) examples (for more details see [4, Ch.6], [16], [17]).
The list of books you might like
