Distributions Theory and Applications
This textbook is an application-oriented introduction to the theory of distributions, a powerful tool used in mathematical analysis. The treatment emphasizes applications that relate distributions to linear partial differential equations and Fourier analy
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Advisory Board Anthony W. Knapp, State University of New York at Stony Brook, Emeritus
J.J. Duistermaat J.A.C. Kolk
Distributions Theory and Applications
Translated from Dutch by J.P. van Braam Houckgeest
J.J. Duistermaat† Mathematical Institute Utrecht University
J.A.C. Kolk Mathematical Institute Utrecht University P.O. Box 80.010 3508 TA Utrecht The Netherlands [email protected]
Translated from Dutch by J.P. van Braam Houckgeest
ISBN 978-0-8176-4672-1 e-ISBN 978-0-8176-4675-2 DOI 10.1007/978-0-8176-4675-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010932757 Mathematics Subject Classification (2010): 46-01, 42-01, 35-01, 28-01, 34-01, 26-01
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To V. S. Varadarajan A True Friend and Source of Inspiration
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Standard Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2
Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4
Differentiation of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5
Convergence of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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