Table Of ContentHARMONIC,
WAVELET AND
p-ADIC ANALYSIS
EDITORIAL BOARD
Nguyh Minh Chuang
Youri V. Egorov
Takeyuki Hida
Andrei Khrennikov
Yves Meyer
David Mumford
Roger Temam
Nguygn Minh Tri
Vii Kim Tudn
HARMONIC,
WAVELET AND
p-ADIC ANALYSIS
editors
N M Choung A Khrennikov
Institute of Mathematics,
Vietnamese-Acad. of Sci. &
Tech., Vietnam Y Meyer
ENS-Cachan, France
Yu V Egorov
University of Toulouse, France D Mumford
Brown University, USA
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HARMONIC, WAVELET AND p-ADIC ANALYSIS
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V
PREFACE
The mutual influence between mathematics, sciences and technology is
more and more widespread. It is both important and interesting to discover
more and more profound connections among different areas of Mathematics,
Sciences and Technology. Particularly exciting has been the discover in
recent years of many relations between harmonic analysis, wavelet analysis
and padic analysis.
So in 2005, from June 10 to 15, at the Quy Nhon University of Vietnam,
an International Summer School on "Harmonic, wavelet and padic analy-
sis" was organized in order to invite a number of well known specialists on
these fields from many countries to give Lectures to teachers, researchers,
and graduate students Vietnamese as well as from foreign institutions.
This volume contains the Lectures given by those invited Professors,
including some from Professors who could not come to the School. These
Lectures are concerned with deterministic as well as stochastic aspects of
the subjects.
The contents of the book are divided in two Parts and four Sections.
Part A deals with wavelets and harmonic analysis. In Section I some
mathematical methods, especially wavelet theory, one of the most powerful
tools for solution of actual problems of mathematical physics and engineer-
ing, are introduced. The connection between wavelet theory and time op-
erators of statistical mechanics is established. Wavelets are also connected
to the theory of stochastic processes. Multiwavelet and multiscale approxi-
mations and localization operator methods are presented.
Section I1 is devoted to some of the most interesting aspects of harmonic
analysis. The nonlinear spectra based on the so called Fiber spectral anal-
ysis with applications are discussed. Here the very famous critical Sobolev
problem is developed, too. The representation theory of affine Hecke al-
gebras, the quantized algebras of functions on affine Hecke algebras are
reviewed and the so called quantized algebras of functions on affine Hecke
algebras of type A and the corresponding q-Schur algebras are defined and
their irreducible unitarizable representation are classified. A survey is made
of the past 40 years of the Andreotti-Grauert legacy as well as its recent de-
vi Preface
velopments (cohomologically q-convex, cohomologically q-complete spaces,
strong q-pseudoconvexity, pseudoconvexity of order m) with some new re-
sults which did not appear elsewhere.
In Part B some recent developments in deterministic and stochastic
analysis over archimedean and non-archimedean fields are introduced. In
Section I11 some Cauchy pseudodifferential problems over padic fields, some
classes of padic Hilbert transformations in some classes of padic spaces,
say BMO, VMO, are investigated. An analogue of probability theory for
probabilities taking values in topological groups is developed. A review is
presented of non-Kolmogorovian models with negative, complex, and padic
probabilities with some applications in physics and cognitive sciences.
Section IV is devoted to archimedean stochastic analysis, more precisely
to some recent aspects on stochastic integral equations of Fredholm type, on
reflecting stochastic differential equations with jumps, on analytic processes
and Levy processes. Here an interesting relation between harmonic analysis,
group theory and white noise theory is also developed.
The Editors
vii
CONTENTS
Preface V
Part A Wavelet and Harmonic Analysis
Chapter I Wavelet and Expectations
$1.W avelets and Expectations: A Different Path to Wavelets 5
Karl Gustafson
$2. Construction of Univariate and Bivariate Exponential Splines 23
Xiaoyan Liu
53. Multiwavelets: Some Approximation-Theoretic Properties,
Sampling on the Interval, and Translation Invariance 37
Peter R. Massopust
$4.M ulti-Scale Approximation Schemes in Electronic Structure
Calculation 59
Reinhold Schneider and Toralf Weber
55. Localization Operators and Time-Frequency Analysis 83
Elena Cordero, Karlheinz Grochenig and Luigi Rodino
Chapter I1 Harmonic Analysis
56. On Multiple Solutions for Elliptic Boundary Value Problem
with Two Critical Exponents 113
Yu. V. Egorov and Yavdat Il’yasov
...
viii Contents
$7. On Calculation of the Bifurcations by the Fibering Approach 141
Yavdat I1 'yasov
$8. On a Free Boundary Transmission Problem for
Nonhomogeneous Fluids 157
Bu.i An Ton
59. Sampling in Paley-Wiener and Hardy Spaces 175
Vu Kim Tuan and Amin Boumenir
$10. Quantized Algebras of Functions on Affine Hecke Algebras 211
Do Ngoc Diep
$11. On the C-Analytic Geometry of q-Convex Spaces 229
Vo Van Tan
Part B P-adic and Stochastic Analysis
Chapter I11 Over padic Field
512. Harmonic Analysis over padic Field I. Some Equations
and Singular Integral Operators 271
Nguyen Manh Chuong, Nguyen Van Co and Le Quang Thuan
$13. p-adic and Group Valued Probabilities 29 1
Andrei Khrennikov
Chapter IV Archimedean Stochastic Analysis
$14. Infinite Dimensional Harmonic Analysis from the Viewpoint
of White Noise Theory 313
Takeyuki Hida
$15. Stochastic Integral Equations of Fredholm Type 331
Shigeyoshi Ogawa
Contents ix
$16. BSDEs with Jumps and with Quadratic Growth Coefficients
and Optimal Consumption 343
Situ Rong
$17. Insider Problems for Markets Driven by LBvy Processes 363
Arturo Kohatsu-Hzga and Makato Yamazato
