Spaces of Approximating Functions with Haar-like Conditions
Tchebycheff (or Haar) and weak Tchebycheff spaces play a central role when considering problems of best approximation from finite dimensional spaces. The aim of this book is to introduce Haar-like spaces, which are Haar and weak Tchebycheff spaces under s
- PDF / 8,662,194 Bytes
- 119 Pages / 461 x 684 pts Page_size
- 86 Downloads / 143 Views
1576
Kazuaki Kitahara
Spaces of Approximating Functions with Haar-like Conditions
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest
Author Kazuaki Kitahara Department of Mathematics Faculty of Education Saga University Saga 840, Japan
Mathematics Subject Classification (1991): 41A50, 41A30
ISBN 3-540-57974-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57974-5 Springer-Verlag New York Berlin Heidelberg CIP-Data applied for 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, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions ofthe German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany
Typesetting: Camera-ready by author/editor SPIN: 10130001 46/3140-543210 - Printed on acid-free paper
To Junko and Toshiya
Preface
Let E be a function space with a norm 11·11 and let G be a finite dimensional subspace of E. Then it is one of the principal themes in approximation theory to study the following problems: For each lEE, find
j
E G such that Ea(f) =
III - ill
= inf gEa
III
gil
and estimate Ea(f). G is called an approximating space and
i is said to be a best approximation to I from G.
If G is chosen in the manner so that Ea(f) is as small as possible and so that functions in G are easy to handle, then G is a good approximating space. For example, in era, b] (=the space of all real-valued continuous functions on [a, b]) with the supremum norm, spaces of polynomials with degree at most n and spaces of continuous and piecewise linear functions with fixed knots are suitable for good approximating spaces. Cebysev (or Haar) spaces and weak Cebysev spaces are generalizations of these two spaces and play a central part when considering the above problems. In fact, properties, characterizations and generalizations of Cebysev and weak Cebysev spaces have been deeply studied during this century. Now, the theory of these spaces has matured. In this book, as approximating spaces, we shall introduce Haar-like spaces, which are Haar and weak Cebysev spaces under special conditions. And we shall study topics of subclasses of Haar-Iike spaces rather than general properties of Haar-like spaces, that is, classes of Cebysev or weak Cebysev spaces, spaces of vector-valued monotone increasing or convex functions and spaces of step functions. Contents are mostly new results and rewritings of the following papers, 13, 14, 15, 16, 17 (Chapter 2),7,8, 9(Chapter 3), 17, 18 (Chapter 4),4,5 (Chapter 5), 2 (Appendix 1), where each number is its reference number. In Chapter 1, Haar-like spaces are defined
Data Loading...
