b2530 International Strategic Relations and China’s National Security: World at the Crossroads TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk b2530_FM.indd 6 01-Sep-16 11:03:06 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE 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. All rights reserved. No reproduction and distribution without permission. Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore Suqi - 10292 - Harmonic Analysis.indd 1 14-07-17 5:01:22 PM 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