Difference Spaces and Invariant Linear Forms
Difference spaces arise by taking sums of finite or fractional differences. Linear forms which vanish identically on such a space are invariant in a corresponding sense. The difference spaces of L2 (Rn) are Hilbert spaces whose functions are characterized
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen
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Rodney Nillsen
Difference Spaces and Invariant Linear Forms
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Author Rodney Nillsen Department of Mathematics University of Wollongong Northfields Avenue Wollongong NSW 2522, Australia E-Mail: [email protected]
The figure on the cover illustrates the graphs of the following functions: the first derivative -2:J:e-%' of e-%'; the first order difference e-(r-l)' - e-(%+!J' of e-(%-l)'; the second derivative 2(2.r 2 - 1 )e-%' of e-%'; and the second order difference e-%' - 2- 1(e-(%-I)' + e-l%+ll') of e-%'. The general relationship between such functions is described in Corollary 3.7 of Chapter III.
Mathematics Subject Classification (1991): 43-02,42-02, 42BIO, 42B20, 43A15, 47F05
ISBN 3-540-58323-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-58323-8 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 of the 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
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.. That is not said right," said the Caterpillar. .. Not quite right, I'm afraid," said Alice, timidly; .. some of the words have got altered." LEWIS CARROLL, in Alice:SO Adventures in Wonderland
PREFACE
The ideas in this work have had a long period of gestation, although the fuller development and expression of them has occurred over a relatively short and intense period. The dominant ideas emerged in an endeavour to answer the following question: if f is a function in L 2 (JR n ), how can the behaviour of its Fourier transform near the origin of JRn be described and characterized? The corresponding question for the circle group had been given a satisfactory answer by Gary Meisters and Wolfgang Schmidt in 1972, so the work also can be regarded as arising from an attempt to extend their result from the compact case of the circle group to the non-compact case of JRn. The answers presented to these and related questions have implications for other areas of analysis; the notable ones being the ranges of partial differential operators, and the behaviour of some of the singular integral operators of classical analysis. I have received a great deal of help and encouragment from many individuals, many of whom are probably not aware of the positive effect they
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