Harmonic Analysis and Fractal Analysis Over Local Fields and Applications

328 Pages·2017·9.148 MB·English

by  Weiyi Su

b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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Published by
World Scientific Publishing Co. Pte. Ltd.
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Library of Congress Cataloging-in-Publication Data
Names: Su, Weiyi.
Title: Harmonic analysis and fractal analysis over local fields and applications /
by Weiyi Su (Nanjing University, China).
Description: New Jersey : World Scientific, 2016. | Includes bibliographical references and index.
Identifiers: LCCN 2016052212 | ISBN 9789813200494 (hardcover : alk. paper)
Subjects: LCSH: Local fields (Algebra) | Algebraic fields. | Harmonic analysis. | Fractal analysis.
Classification: LCC QA247 .S87 2016 | DDC 512.7--dc23
LC record available at https://lccn.loc.gov/2016052212
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Harmonic Analysis and Fractal Analysis over Local Fields and Applications
@ Weiyi Su
The Work is originally published by Science Press in 2011.
This edition is published by World Scientific Publishing Company Pte Ltd by arrangement with
Science Press, Beijing, China.
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August8,2017 17:20 HarmonicAnalysisandFractalAnalysis(cid:21)10292 9789813200494 pagev
Preface
1975, M. H. Taibleson[100] published a consummate book Fourier Analysis
on Local Fields in which he has given a nice summary of research on local
(cid:28)elds, and it formed the foundation of harmonic analysis on local (cid:28)elds.
(cid:16)Harmonicanalysisonlocal(cid:28)elds(cid:17) isoneofimportantbranchesofabstract
harmonic analysis, one may take local (cid:28)eld as underlying space to study
lots of interesting and important topics in the area of abstract harmonic
analysis, including many hotspots in theory and applications.
Almost at the same period, B. B. Mandelbrot[36],[37] introduced a quite
new concept (cid:16)fractal(cid:17), in fact, he has explored a new area, and promoted
formation of a new bright spot of the geometric measure theory. The sub-
jects in fractal geometry are those, called (cid:16)fractal(cid:17) by Mandelbrot, which
often appear in modern science areas, or industry application areas, but
could not be dealt by classical mathematical methods. We may mention
someofthem: functionsarenotdi(cid:27)erentiable,orevennotcontinuous,such
astheWeierstrassfunction,Cantorfunction,...; setsarenotregular,such
as Koch curve, Cantor set, .... These functions and sets appear in many
scienti(cid:28)c areas, such as, physics, chemistry, astronomy, computer science,
lifescience,clinicalmedicinescience,geology,meteorology,aswellassignal
analysis and transfer in some industry applications. Then scientists accept
theidea(cid:16)fractal(cid:17) veryquickly,andtheyhaverecognizedthat(cid:16)fractal(cid:17) isone
oftheessentialpropertiesofnon-linearphenomena. Sothatthe(cid:16)non-linear
science(cid:17) has appeared in 90s of last century.
Alocal(cid:28)eldisalocallycompact,totallydisconnected,non-Archimedean
valued, complete topological (cid:28)eld, it has extensive application prospects,
for instance, the dyadic numbers and p-adic numbers are used widely, and
v
August8,2017 17:20 HarmonicAnalysisandFractalAnalysis(cid:21)10292 9789813200494 pagevi
vi Harmonic Analysis and Fractal Analysis over Local Fields and Applications
indeed they are special cases of local (cid:28)elds. More interesting fact is that
lotsoffractalsetscanbedescribedassetsinlocal(cid:28)elds. Also,somefractal
functions can be described as functions de(cid:28)ned on local (cid:28)elds.
Now a new cross-scienti(cid:28)c area between pure mathematics and applied
mathematics emerges combining (cid:16)harmonic analysis on local (cid:28)elds(cid:17) with
(cid:16)fractal analysis(cid:17). This new cross-scienti(cid:28)c branch reveals one of essential
properties of our universe (cid:22) non-linearity, and becomes one of the main
topics of non-linear science.
Meanwhile, a lager number new problems have appeared in life science
andclinicalmedicinesciencearea, forinstance, whatkindsofhumangenes
control hearts of persons, and what kinds of genes control their lungs, con-
trol their livers, ...? And what role does each gene play? More interesting
andsigni(cid:28)cantproblemis: canwedeterminethemalignancyoflivercancer
exactlybytheboundaryshapeoflesionofcancer,orbyfractaldimensionof
boundary? Manyresearchresultsshowthatlotsofnewproblemsappeared
inthevariousscienceareascanbedescribedbyvirtueofestablishingmath-
ematical models on local (cid:28)elds, and use the tools of local (cid:28)elds to (cid:28)nd the
corresponding solutions.
The propose for writing this book is to exhibit a new research direction
(cid:22)(cid:16)harmonicanalysisandfractalanalysisoverlocal(cid:28)eldsandapplications(cid:17),
includingfundamentalsoflocal(cid:28)eldsandfractals,andnewresearchresults
inmorethan30yearssince1980s. Fromthemathematicsfoundationtheory
to the applications of clinical medicine science. It forms a self-contained
body, and shows that in this new cross research area what has been done,
what researchers are doing now, and what will be done in the future.
The book is divided as 3 parts and 7 chapters. The (cid:28)rst part (chapter
1 and 2) consists fundamentals of local (cid:28)elds. The second part (chapter 3
and 4) is the foundation theory of harmonic analysis over local (cid:28)elds. The
third part (chapter 5, 6, and 7) is the theory and applications of fractal
analysis over local (cid:28)elds.
In Chapter 1, the basic knowledge of Galois (cid:28)eld GF (p) and the struc-
tures of local (cid:28)elds are introduced. In Chapter 2, the character groups of
local (cid:28)elds are exhibited, and some important formulas about local (cid:28)elds
andtheircharactergroupsareevaluated. Chapter3andChapter4contain
the foundation theory of harmonic analysis: Fourier analysis and function
spaces on local (cid:28)elds. Specially, the new calculus based on local (cid:28)elds as
August8,2017 17:20 HarmonicAnalysisandFractalAnalysis(cid:21)10292 9789813200494 pagevii
Preface vii
underlyingspacesisestablished,thep-typederivativesandp-typeintegrals
are de(cid:28)ned by virtue of so called pseudo-di(cid:27)erential operators, and impor-
tant properties of p-type calculus are proved. Then, a comparison between
Euclidean spaces and local (cid:28)elds is given. Chapter 5 discusses the fractal
analysis over local (cid:28)elds, including basic knowledge of fractals. Some frac-
tal sets and fractal functions are constructed on local (cid:28)elds, for example,
Cantor type sets, Weierstrass type functions, and so on. Then the fea-
tures of analysis on local (cid:28)elds and that on Euclidean spaces are pointed
out. Chapter 6 devotes to the topic on fractal partial di(cid:27)erential equations
(cid:22) a challenging research topic in fractal analysis. Finally, in Chapter 7,
problems and applications in clinical medicine science are shown.
Comparing with classical harmonic analysis on Euclidean spaces, the
harmonic analysis and fractal analysis over local (cid:28)elds are quite young,
only more than 30 years, it is just a moment in history, a drop in ocean of
knowledge. However, it plays an important role to reveal the essences of
universe, life regularities of living beings, and becomes one of the cores
of non-linear science. We may expect that the harmonic analysis and
fractal analysis on Euclidean spaces and over local (cid:28)elds, both will be
powerful nice tools for humankinds to recognize and study universe and
nature.
I summarize and cite the research results in the period of more than
30 years of the group of harmonic analysis and fractal analysis in the De-
partment of Mathematics, Nanjing University, obtained by my colleagues
Professors Weixing Zheng, Zelin He, Huikun Jiang, Zhaoxi Wang, Zhaojin
Wu and Ph.D students Huojun Ruan, Kui Yao, Hua Qiu, Yueping Zhu,
Shijun Zheng, Guangcai Zhou, Ning Xu, Bo Wu, Baoyi Wu, YinLi, Lintao
Ma; aswellasundergraduatestudentsYunPeng, KaimingShen, Qingsong
Gu, et al. It is my pleasure to work with them. I would like to express my
heartfelt thanks to all of them for pleasant co-working.
IamgreatlyindebtedtoProfessorSileiWanginZhejiangUniversityand
Professor Shenzhen Lu in Beijing Normal University for their abundant
help. Moreover, I would like to thank Professor Yudong Qiu of Nanjing
Gulou Hospital and his graduate students for fruitful cooperation.
Special thanks due to the editors of Science Press in Beijing, especially,
the chief editor Yanchao Zhao for the e(cid:27)ort during the publication of this
book.
August8,2017 17:20 HarmonicAnalysisandFractalAnalysis(cid:21)10292 9789813200494 pageviii
viii Harmonic Analysis and Fractal Analysis over Local Fields and Applications
I try to show readers a development of a new science area, and its
splendid structure, nice theory and applications based on strict mathemat-
ics theory. Meanwhile to make smooth reading for readers, I have tried to
organize the contents in order of di(cid:30)culty, to write in a pithy style, and to
make the key point stand out. Hope these e(cid:27)orts can bene(cid:28)t newer in the
area.
De(cid:28)nitelytherearepossiblemistakesandshort-comingsinsuchabook
about so rapidly developing research area. I hope experts and readers can
kindlyfeedbackwithvaluedinstructions. Allsuggestionsandcriticismsare
welcome.
Weiyi Su
Nanjing University, Nanjing, China
December 2016
August8,2017 17:20 HarmonicAnalysisandFractalAnalysis(cid:21)10292 9789813200494 pageix
Contents
Preface v
1. Preliminary 1
1.1 Galois (cid:28)eld GF(p) . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Galois (cid:28)eld GF(p), characteristic number p . . . . 1
1.1.2 Algebraic extension (cid:28)elds of Galois (cid:28)eld GF(p) . . 3
1.2 Structures of local (cid:28)elds . . . . . . . . . . . . . . . . . . . 5
1.2.1 De(cid:28)nitions of local (cid:28)elds . . . . . . . . . . . . . . 5
1.2.2 Valued structure of a local (cid:28)eld K . . . . . . . . 6
q
1.2.3 Haar measure and Haar integral on a local (cid:28)eld K 7
q
1.2.4 Important subsets in a local (cid:28)eld K . . . . . . . . 9
q
1.2.5 Base for neighborhood system of a local (cid:28)eld K . 10
q
1.2.6 Expressions of elements in K and operations . . . 11
q
1.2.7 Important properties of balls in a local (cid:28)eld K . 14
p
1.2.8 Order structure in a local (cid:28)eld K . . . . . . . . . 15
p
1.2.9 Relationship between local (cid:28)eld K and Euclidean
p
space R . . . . . . . . . . . . . . . . . . . . . . . . 17
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2. Character Group Γ of Local Field K 21
p p
2.1 Character groups of locally compact groups . . . . . . . . 21
2.1.1 Characters of groups . . . . . . . . . . . . . . . . 21
2.1.2 Charactersandcharactergroupsoflocallycompact
groups . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.3 Pontryagin dual theorem . . . . . . . . . . . . . . 23
ix