Decay of the Fourier Transform: Analytic and Geometric Aspects

226 Pages·2014·1.662 MB·English

by  Iosevich, Alexander, Liflyand, Elijah

Alex Iosevich
(cid:40)(cid:79)(cid:76)(cid:77)(cid:68)(cid:75)(cid:3)(cid:47)(cid:76)(cid:190)(cid:3)(cid:92)(cid:68)(cid:81)(cid:71)
Decay
of the Fourier
Transform
(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)
(cid:42)(cid:72)(cid:82)(cid:80)(cid:72)(cid:87)(cid:85)(cid:76)(cid:70)(cid:3)(cid:36)(cid:86)(cid:83)(cid:72)(cid:70)(cid:87)(cid:86)

Alex Iosevich • Elijah Liflyand
Decay of the Fourier Transform
Analytic and Geometric Aspects
Alex Iosevich Elijah Liflyand
Department of Mathematics Department of Mathematics
University of Rochester Bar-Ilan University
Rochester, NY, USA Ramat-Gan, Israel
ISBN 978-3-0348-0624-4 ISBN 978-3-0348-0625-1 (eBook)
DOI 10.1007/978-3-0348-0625-1
Springer Basel Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014952035

Mathematics Subject Classification (2010): 42A, 42B

© Springer Basel 2014
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now
known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with
reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed
on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or
parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its
current version, and permission for use must always be obtained from Springer. Permissions for use may be
obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the
respective Copyright Law.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
While the advice and information in this book are believed to be true and accurate at the date of publication,
neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or
omissions that may be made. The publisher makes no warranty, express or implied, with respect to the
material contained herein.

Printed on acid-free paper

Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com)
TO OUR FAMILIES (cid:676)(cid:696)(cid:695)(cid:677)(cid:676)(cid:673)(cid:3)(cid:677)(cid:688)(cid:681)(cid:698)(cid:677)(cid:679)(cid:692)(cid:697)(cid:686)(cid:684)

Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Part 0 Preliminaries
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 Basic Properties of the Fourier Transform
1.1 L1-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 L2-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Poissonsummation formula . . . . . . . . . . . . . . . . . . . . . . . 16
Part 1 Analytic (and Geometric) Aspects
2 Oscillatory Integrals
2.1 The method of stationary phase . . . . . . . . . . . . . . . . . . . . . 19
2.2 Erd´elyi type results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 The Fourier transform on a convex set . . . . . . . . . . . . . . . . . 27
2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.1 Hyperbolic means . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.2 Multiple Fourier integrals . . . . . . . . . . . . . . . . . . . . 38
2.4.3 Generalized Bochner–Riesz means of critical order . . . . . . 42
3 The Fourier Transform of Convex and Oscillating Functions
3.1 Convex functions with singularities . . . . . . . . . . . . . . . . . . . 47
3.2 Asymptotic behavior in a wider sense. . . . . . . . . . . . . . . . . . 51
3.3 Integrability of trigonometric series . . . . . . . . . . . . . . . . . . . 52
3.4 Analogous function spaces . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5 Asymptotics of the Fourier transform . . . . . . . . . . . . . . . . . . 60
3.5.1 Hardy type spaces . . . . . . . . . . . . . . . . . . . . . . . . 60
vii
viii Contents
3.5.2 The Fourier transform of a function with shifted
singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5.3 Amalgam type spaces. . . . . . . . . . . . . . . . . . . . . . . 77
3.6 Applications and further progress . . . . . . . . . . . . . . . . . . . . 81
3.7 Multivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4 The Fourier Transform of a Radial Function
4.1 Fractional derivatives and classes of functions . . . . . . . . . . . . . 94
4.2 Existence of the Fourier transform and Bessel type functions . . . . 99
4.3 Proof of the existence theorem . . . . . . . . . . . . . . . . . . . . . 104
4.4 Passageto a one-dimensional Fourier transform . . . . . . . . . . . . 118
4.5 Certain applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Part 2 Geometric (and Analytic) Aspects
5 L2-average Decay of the Fourier Transform of a
Characteristic Function of a Convex Set
5.1 L2-average decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2 Proof of Theorem 5.5 and Theorem 5.7 . . . . . . . . . . . . . . . . 131
5.2.1 Decomposition of the boundary . . . . . . . . . . . . . . . . . 132
5.2.2 Singular directions . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2.3 Non-singular directions . . . . . . . . . . . . . . . . . . . . . . 133
6 L1-average Decay of the Fourier Transform of a
Characteristic Function of a Convex Set
6.1 Preliminary discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2 A variety of arguments . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.2.1 Lp-averagedecay of the Fourier transform . . . . . . . . . . . 140
6.2.2 Inscribed polygons . . . . . . . . . . . . . . . . . . . . . . . . 141
6.2.3 The image of the Gauss map. . . . . . . . . . . . . . . . . . . 142
6.2.4 Comparing the previous arguments . . . . . . . . . . . . . . . 143
6.2.5 A lower bound for all convex bodies. . . . . . . . . . . . . . . 144
6.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.3.1 Lattice points . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.3.2 Irregularities of distribution . . . . . . . . . . . . . . . . . . . 146
6.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7 Geometry of the Gauss Map and Lattice Points in Convex Domains
7.1 Two main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.2 Examples and preliminary results . . . . . . . . . . . . . . . . . . . . 163
7.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.2.2 Estimates for the Fourier transform . . . . . . . . . . . . . . . 164
Contents ix
7.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.3.1 Proof of Theorem 7.1 . . . . . . . . . . . . . . . . . . . . . . . 165
7.3.2 Proof of Theorem 7.2 . . . . . . . . . . . . . . . . . . . . . . . 167
7.3.3 Proof of Theorem 7.8 . . . . . . . . . . . . . . . . . . . . . . . 168
7.3.4 Proof of Theorem 7.9 . . . . . . . . . . . . . . . . . . . . . . . 170
8 Average Decay Estimates for Fourier Transforms of Measures
Supported on Curves
8.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.2 Upper bounds, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.3 Upper bounds, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.4 Upper bounds, III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.5 Asymptotics for oscillatory integrals revisited . . . . . . . . . . . . . 189
8.6 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221