Discrete Spectral Synthesis and Its Applications
In order to study discrete Abelian groups with wide range applications, the use of classical functional equations, difference and differential equations, polynomial ideals, digital filtering and polynomial hypergroups is required. This book covers several
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László Székelyhidi
Discrete Spectral Synthesis and Its Applications
ABC
László Székelyhidi University of Debrecen Institute of Mathematics H-4010 Debrecen, Hungary email: [email protected]
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 ISBN-13 ISBN-10 ISBN-13
1-4020-4636-7 (HB) 978-1-4020-4636-0 (HB) 1-4020-4637-5 (e-Book) 978-1-4020-4637-7 (e-Book)
Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com
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To my sons and my wife
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
Spectral synthesis and spectral analysis . . . . . . . . . . . . . . . . . . . . 7 2.1 The basic problems of spectral analysis and spectral synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Spectral analysis and synthesis on L1 (G) . . . . . . . . . . . . . . . . . . . 8 2.3 Spectral analysis and synthesis on L∞ (G) . . . . . . . . . . . . . . . . . . 11 2.4 Spectral analysis and synthesis on C(G) . . . . . . . . . . . . . . . . . . . . 17
3
Spectral analysis and spectral synthesis on discrete Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Spectral analysis on discrete Abelian torsion groups . . . . . . . . . . 3.2 Spectral analysis on Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Spectral analysis on commutative semigroups . . . . . . . . . . . . . . . 3.4 Spectral synthesis and polynomial ideals . . . . . . . . . . . . . . . . . . . . 3.5 The failure of spectral synthesis on some types of discrete Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Spectral synthesis on Abelian torsion groups . . . . . . . . . . . . . . . . 3.7 Polynomial functions and spectral synthesis . . . . . . . . . . . . . . . . .
25 25 27 27 29 34 38 42
4
Spectral synthesis and functional equations . . . . . . . . . . . . . . . . 4.1 Convolution type functional equations . . . . . . . . . . . . . . . . . . . . . . 4.2 Mean value type functional equations . . . . . . . . . . . . . . . . . . . . . . 4.3 A functional equation in digital filtering . . . . . . . . . . . . . . . . . . . .
49 49 52 60
5
Mean periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Fourier transform of mean peri
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