CONTENTS -- Preface -- Part A Wavelet and Harmonic Analysis -- Chapter I Wavelet and Expectations -- 1. Wavelets and Expectations: A Different Path to Wavelets Karl Gustafson -- 1. Introduction, Background, and Summary -- 2. 1956: Wavelets and Seismic Oil Prospecting -- 3. 1976: Quantum Mechanics and Stochastic Processes -- 4. 1980: Higher Dimensional Spectral Estimation -- 5. 1985: Kolmogorov Automorphisms as Wavelet Refinements -- 6. 1991: Wavelets as Wandering Subspaces for Time Operators -- 7. 1998: Haar and Wavelet Bases and Differential Equations -- 8. 2005: Positivity in Kolmogorov and Wavelet Structures -- 9. Recent Related Work by Others -- 10. Conclusions -- Acknowledgements -- References -- 2. Construction of Univariate and Bivariate Exponential Splines Xiaoyan Liu -- 1. Introduction -- 2. Integral Iteration Formulas for Constructing Univariate Exponential Splines -- 3. Properties of Exponential Splines -- 4. Hyperbolic Splines -- 5. A Univariate Orthonormal Exponential Spline with Minimum Support -- 6. Integral Iteration Formulas for Constructing Bivariate Exponential Splines -- References -- 3. Multiwavelets: Some Approximation-Theoretic Properties, Sampling on the Interval, and Translation Invariance Peter R. Massopust -- 1. Introduction -- 2. Notation and Preliminaries -- 2.1. Shift-invariant spaces -- 2.2. Refinable spaces -- 2.3. Multiwavelets -- 2.4. Reconstruction and decomposition algorithm -- 2.5. Stability of projectors -- 2.6. Approximation order and smoothness -- 3. Sampling with Multiwavelets -- 3.1. The GHM scaling vector and DGHM multiwavelet -- 4. Multiwavelets on the Interval -- 4.1. Function sampling on [0,1] -- 5. Translation Invariance -- 5.1. Matrix-valued refinement -- 5.2. Regularity and moments -- 5.3. Coiflet property -- 5.4. An example: The DGHM multiwavelet -- References.

4. Multi-Scale Approximation Schemes in Electronic Structure Calculation Reinhold Schneider and Toralf Weber -- 1. Introduction -- 2. Electronic Structure Calculation -- 3. Effective One-Particle Models -- 3.1. Hartree-Fock equations -- 3.2. Density functional theory -- 3.3. Self-consistent field approximation -- 3.4. Projection methods -- 3.5. Multiresolution spaces -- 4. Projection Methods for Eigenvalue Problems of Schrodinger Type -- 5. Numerical Experiments -- References -- 5. Localization Operators and Time-Frequency Analysis Elena Cordero, Karlheinz Grochenig and Luigi Rodino -- 1. Introduction and Definitions -- 2. Time-Frequency Methods -- 3. Function Spaces -- 4. Regularity Results -- 4.1. Sufficient conditions for boundedness and schatten class -- 4.2. Ultra-distributions with compact support as symbols -- 4.3. Compactness of localization operators -- 5. Necessary Conditions -- 6. Composition Formula -- 6.1. Exact product -- 6.2. Asymptotic product -- 7. Multilinear Localization Operators -- References -- Chapter II Harmonic Analysis -- 6. On Multiple Solutions for Elliptic Boundary Value Problem with Two Critical Exponents Yu. V. Egorov and Yavdat Il'yasov -- 1. Introduction and Main Results -- 2. The Basic Variational Problems -- 2.1. The level of ground state and the Sobolev level -- 3. Existence of the Ground State -- 4. The Palais-Smale Property -- 5. Subcritical Auxiliary Results -- 6. Proofs of Theorems 1.1, 1.2 -- 6.1. Proof of statement 1) of Theorem 1.1 -- 6.2. End of the Proofs of Propositions 2.2) in Theorem 1.1 and 2) in Theorem 1.2 -- References -- 7. On Calculation of the Bifurcations by the Fibering Approach Yavdat I1 'yasov -- 1. Introduction -- 2. Fiber Spectral Analysis -- 3. The Problem with Inhomogeneous Indefinite Nonlinearit ies -- 4. A Problem with Concave-Convex Nonlinearity.

5. The Equations with p&q- Laplacian -- 6. Solutions of Minimal Period for a Hamatonian System with Potential Indefinite in Sign -- References -- 8. On a Free Boundary Transmission Problem for Nonhomogeneous Fluids Bu.i An Ton -- 1. Introduction -- 2. Notations, assumptions, preliminary results -- 3. The initial value problem (2) -- 4. A linear transmission problem in non cylindrical domains -- 5. The nonlinear case -- References -- 9. Sampling in Paley-Wiener and Hardy Spaces Vu Kim Tuan and Amin Boumenir -- 1. Sampling in Paley-Wiener Spaces -- 1.1. Inverse spectral problem -- 1.2. Suficient conditions for spectral functions -- 1.3. Sampling formula for PW1I2 -- 1.4. Sampling formula f o r bandlimited signals -- 1.5. Error analysis -- 2. Sampling in Hardy Space -- 2.1. Sampling formula -- 2.2. Truncation error -- 2.3. Example -- References -- 10. Quantized Algebras of Functions on Affine Hecke Algebras Do Ngoc Diep -- Introduction -- 1. Definition of the Quantized Algebras of Functions -- 1.1. -- 1.1.1. p-adic presentation -- 1.1.2. A B n e heclce algebras W(v, W&) -- 1.2. -- 1.2.1. Admissible representations of p-adic groups -- 1.2.2. Dipper-James construction of irreducible finite dimensional representations of W(v, W&) -- 1.2.3. The langlands correspondence -- 1.3. -- 1.3.1. Quantized algebras of functions -- 1.3.2. Inclusion -- 2. Irreducible Representations -- 2.1. Rank I case -- 2.2. Rank r case -- 3. Schur-Weyl Duality -- 3.1. -- 3.1.1. v-Schur algebras Sn,,(v) -- 3.1.2. v-Schur duality -- 3.2. -- 3.2.1. Restriction maps -- 3.2.2. Description of irreducible representations -- Acknowledgments -- References -- 11. On the C-Analytic Geometry of q-Convex Spaces Vo Van Tan -- 1. Introduction -- 2. The Andreotti Grauert Legacy -- 2.1. The initial challenge -- 2.2. The prime time -- 2.3. The shattered dreams -- 2.4. The watchtower.

2.5. The bottom line -- 3. The Holomorph-convex Spaces -- 3.1. Preliminaries -- 3.2. The compactness -- 3.3. The cohomologically q-convexity -- 3.4. The q-convexity -- 4. The K-complete Spaces -- 4.1. Preliminaries -- 4.2. The twin primes -- 4.3. An indentation -- 5. The Cultural Diversity -- 5.1. The GAGA duality -- 5.2. Pluribus unum -- 6. Epilogue -- Acknowledgements -- References -- Part B P-adic and Stochastic Analysis -- Chapter III Over padic Field -- 13. p-adic and Group Valued Probabilities Andrei Khrennikov -- 1. Historical Introduction -- 2. p-adic Lessons -- 2.1. p-adic numbers -- 2.2. p-adic frequency probability model -- 2.3. Measure-theoretic approach -- 3. padic Limit Theorems -- 3.1. p-adic asymptotics of bernoulli probabilities -- 3.2. Laws of large numbers -- 3.3. The central limit theorem -- 4. Axiomatics for Probability Valued in a Topological Group -- 5. Interpretation of Probabilities with Values in a Topological Group, Statistics -- Acknowledgments -- References -- 12. Harmonic Analysis over padic Field I. Some Equations and Singular Integral Operators Nguyen Manh Chuong, Nguyen Van Co and Le Quang Thuan -- 1. Introduction -- 2. Preliminaries -- 3. A p-adic Cauchy Problem -- 3.1. The problem and some spectral properties -- 3.2. Some results -- 4. The p-adic Hilbert Transform -- 5. The Boundedness in padic Space Lq -- 6. The Boundedness the p-adic Space BMO (Q -- ) -- 7. The Boundedness in the padic Space VMO(QF) -- 8. The Boundedness in the p-adic Space Lq(Q -- , w ( z ) , l k ) -- Acknowledgment -- References -- Chapter IV Archimedean Stochastic Analysis -- 14. Infinite Dimensional Harmonic Analysis from the Viewpoint of White Noise Theory Takeyuki Hida -- 1. Introduction -- 2. Infinite Dimensional Rotation Group O(E) -- 3. The LQvy Group, the Windmill Subgroup and the Sign-Changing Subgroup of O(E) -- 4. Whiskers.

Unitary representation of the group O(E) -- 5. Infinite Dimensional Unitary Group -- Acknowledgements -- 15. Stochastic Integral Equations of Fredholm Type Shigeyoshi Ogawa -- 1. Introduction -- 2. Uni-dimensional Case -- 2.1. Assumptions and notations -- 2.2. Simple case -- 2.3. General case -- 3. Multi Dimensional Case -- 3.1. Stochastic Fourier transformation -- 3.2. Results -- References -- 16. BSDEs with Jumps and with Quadratic Growth Coefficients and Optimal Consumption Situ Rong -- 1. Conception -- 2. A Comparison Theorem and An Approximation Lemma -- 3. Existence of Solution for BSDE with Quadratic Growth in q -- 4. Uniqueness of Solution to BSDE with Quadratic Growth in q -- 5. Existence of Solution for BSDE with Quadratic Growth in q and y -- 6. Uniqueness of Solutions for BSDE with Jumps and with Quadratic Growth in q and y -- 7. Optimal Consumption -- References -- 17. Insider Problems for Markets Driven by LBvy Processes Arturo Kohatsu-Hzga and Makato Yamazato -- 1. Introduction -- 2. Enlargement of Filtration with Respect to Semimart ingales -- 3. Optimal Portfolios for Insiders -- 4. Examples of Simple Compound Poisson Processes -- References.