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This book discusses some aspects of the theory of partial differential equations from the viewpoint of probability theory. It is intended not only for specialists in partial differential equations or probability theory but also for specialists in asymptotic methods and in functional analysis. It is also of interest to physicists who use functional integrals in their research. The work contains results that have not previously appeared in book form, including research contributions of the author.

Partial differential equations --- Differential equations, Partial. --- Probabilities. --- Integration, Functional. --- Functional integration --- Functional analysis --- Integrals, Generalized --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- A priori estimate. --- Absolute continuity. --- Almost surely. --- Analytic continuation. --- Axiom. --- Big O notation. --- Boundary (topology). --- Boundary value problem. --- Bounded function. --- Calculation. --- Cauchy problem. --- Central limit theorem. --- Characteristic function (probability theory). --- Chebyshev's inequality. --- Coefficient. --- Comparison theorem. --- Continuous function (set theory). --- Continuous function. --- Convergence of random variables. --- Cylinder set. --- Degeneracy (mathematics). --- Derivative. --- Differential equation. --- Differential operator. --- Diffusion equation. --- Diffusion process. --- Dimension (vector space). --- Direct method in the calculus of variations. --- Dirichlet boundary condition. --- Dirichlet problem. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Elliptic partial differential equation. --- Equation. --- Existence theorem. --- Exponential function. --- Feynman–Kac formula. --- Fokker–Planck equation. --- Function space. --- Functional analysis. --- Fundamental solution. --- Gaussian measure. --- Girsanov theorem. --- Hessian matrix. --- Hölder condition. --- Independence (probability theory). --- Integral curve. --- Integral equation. --- Invariant measure. --- Iterated logarithm. --- Itô's lemma. --- Joint probability distribution. --- Laplace operator. --- Laplace's equation. --- Lebesgue measure. --- Limit (mathematics). --- Limit cycle. --- Limit point. --- Linear differential equation. --- Linear map. --- Lipschitz continuity. --- Markov chain. --- Markov process. --- Markov property. --- Maximum principle. --- Mean value theorem. --- Measure (mathematics). --- Modulus of continuity. --- Moment (mathematics). --- Monotonic function. --- Navier–Stokes equations. --- Nonlinear system. --- Ordinary differential equation. --- Parameter. --- Partial differential equation. --- Periodic function. --- Poisson kernel. --- Probabilistic method. --- Probability space. --- Probability theory. --- Probability. --- Random function. --- Regularization (mathematics). --- Schrödinger equation. --- Self-adjoint operator. --- Sign (mathematics). --- Simultaneous equations. --- Smoothness. --- State-space representation. --- Stochastic calculus. --- Stochastic differential equation. --- Stochastic. --- Support (mathematics). --- Theorem. --- Theory. --- Uniqueness theorem. --- Variable (mathematics). --- Weak convergence (Hilbert space). --- Wiener process.

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This book describes the theory and applications of discrete orthogonal polynomials--polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.

Orthogonal polynomials --- Asymptotic theory --- Orthogonal polynomials -- Asymptotic theory. --- Polynomials. --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Asymptotic theory. --- Asymptotic theory of orthogonal polynomials --- Algebra --- Airy function. --- Analytic continuation. --- Analytic function. --- Ansatz. --- Approximation error. --- Approximation theory. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Asymptotic formula. --- Beta function. --- Boundary value problem. --- Calculation. --- Cauchy's integral formula. --- Cauchy–Riemann equations. --- Change of variables. --- Complex number. --- Complex plane. --- Correlation function. --- Degeneracy (mathematics). --- Determinant. --- Diagram (category theory). --- Discrete measure. --- Distribution function. --- Eigenvalues and eigenvectors. --- Equation. --- Estimation. --- Existential quantification. --- Explicit formulae (L-function). --- Factorization. --- Fredholm determinant. --- Functional derivative. --- Gamma function. --- Gradient descent. --- Harmonic analysis. --- Hermitian matrix. --- Homotopy. --- Hypergeometric function. --- I0. --- Identity matrix. --- Inequality (mathematics). --- Integrable system. --- Invariant measure. --- Inverse scattering transform. --- Invertible matrix. --- Jacobi matrix. --- Joint probability distribution. --- Lagrange multiplier. --- Lax equivalence theorem. --- Limit (mathematics). --- Linear programming. --- Lipschitz continuity. --- Matrix function. --- Maxima and minima. --- Monic polynomial. --- Monotonic function. --- Morera's theorem. --- Neumann series. --- Number line. --- Orthogonal polynomials. --- Orthogonality. --- Orthogonalization. --- Parameter. --- Parametrix. --- Pauli matrices. --- Pointwise convergence. --- Pointwise. --- Polynomial. --- Potential theory. --- Probability distribution. --- Probability measure. --- Probability theory. --- Probability. --- Proportionality (mathematics). --- Quantity. --- Random matrix. --- Random variable. --- Rate of convergence. --- Rectangle. --- Rhombus. --- Riemann surface. --- Special case. --- Spectral theory. --- Statistic. --- Subset. --- Theorem. --- Toda lattice. --- Trace (linear algebra). --- Trace class. --- Transition point. --- Triangular matrix. --- Trigonometric functions. --- Uniform continuity. --- Unit vector. --- Upper and lower bounds. --- Upper half-plane. --- Variational inequality. --- Weak solution. --- Weight function. --- Wishart distribution. --- Orthogonal polynomials - Asymptotic theory

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This book has as its subject the boundary value theory of holomorphic functions in several complex variables, a topic that is just now coming to the forefront of mathematical analysis. For one variable, the topic is classical and rather well understood. In several variables, the necessary understanding of holomorphic functions via partial differential equations has a recent origin, and Professor Stein's book, which emphasizes the potential-theoretic aspects of the boundary value problem, should become the standard work in the field.Originally published in 1972.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Mathematical potential theory --- Holomorphic functions --- Harmonic functions --- Holomorphic functions. --- Harmonic functions. --- Fonctions de plusieurs variables complexes. --- Functions of several complex variables --- Functions, Harmonic --- Laplace's equations --- Bessel functions --- Differential equations, Partial --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Functions, Holomorphic --- Absolute continuity. --- Absolute value. --- Addition. --- Ambient space. --- Analytic function. --- Arbitrarily large. --- Bergman metric. --- Borel measure. --- Boundary (topology). --- Boundary value problem. --- Bounded set (topological vector space). --- Boundedness. --- Brownian motion. --- Calculation. --- Change of variables. --- Characteristic function (probability theory). --- Combination. --- Compact space. --- Complex analysis. --- Complex conjugate. --- Computation. --- Conformal map. --- Constant term. --- Continuous function. --- Coordinate system. --- Corollary. --- Cramer's rule. --- Determinant. --- Diameter. --- Dimension. --- Elliptic operator. --- Estimation. --- Existential quantification. --- Explicit formulae (L-function). --- Exterior (topology). --- Fatou's theorem. --- Function space. --- Green's function. --- Green's theorem. --- Haar measure. --- Half-space (geometry). --- Harmonic function. --- Hilbert space. --- Holomorphic function. --- Hyperbolic space. --- Hypersurface. --- Hölder's inequality. --- Invariant measure. --- Invertible matrix. --- Jacobian matrix and determinant. --- Line segment. --- Linear map. --- Lipschitz continuity. --- Local coordinates. --- Logarithm. --- Majorization. --- Matrix (mathematics). --- Maximal function. --- Measure (mathematics). --- Minimum distance. --- Natural number. --- Normal (geometry). --- Open set. --- Order of magnitude. --- Orthogonal complement. --- Orthonormal basis. --- Parameter. --- Poisson kernel. --- Positive-definite matrix. --- Potential theory. --- Projection (linear algebra). --- Quadratic form. --- Quantity. --- Real structure. --- Requirement. --- Scientific notation. --- Sesquilinear form. --- Several complex variables. --- Sign (mathematics). --- Smoothness. --- Subgroup. --- Subharmonic function. --- Subsequence. --- Subset. --- Summation. --- Tangent space. --- Theorem. --- Theory. --- Total variation. --- Transitive relation. --- Transitivity. --- Transpose. --- Two-form. --- Unit sphere. --- Unitary matrix. --- Vector field. --- Vector space. --- Volume element. --- Weak topology.

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This volume investigates the interplay between the classical theory of automorphic forms and the modern theory of representations of adele groups. Interpreting important recent contributions of Jacquet and Langlands, the author presents new and previously inaccessible results, and systematically develops explicit consequences and connections with the classical theory. The underlying theme is the decomposition of the regular representation of the adele group of GL(2). A detailed proof of the celebrated trace formula of Selberg is included, with a discussion of the possible range of applicability of this formula. Throughout the work the author emphasizes new examples and problems that remain open within the general theory.TABLE OF CONTENTS: 1. The Classical Theory 2. Automorphic Forms and the Decomposition of L2(PSL(2,R) 3. Automorphic Forms as Functions on the Adele Group of GL(2) 4. The Representations of GL(2) over Local and Global Fields 5. Cusp Forms and Representations of the Adele Group of GL(2) 6. Hecke Theory for GL(2) 7. The Construction of a Special Class of Automorphic Forms 8. Eisenstein Series and the Continuous Spectrum 9. The Trace Formula for GL(2) 10. Automorphic Forms on a Quaternion Algebr?

Number theory --- Representations of groups --- Linear algebraic groups --- Adeles --- Representations of groups. --- Automorphic forms. --- Linear algebraic groups. --- Adeles. --- Nombres, Théorie des --- Formes automorphes --- Automorphic forms --- Algebraic fields --- Algebraic groups, Linear --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- Automorphic functions --- Forms (Mathematics) --- Group representation (Mathematics) --- Groups, Representation theory of --- Nombres, Théorie des. --- Abelian extension. --- Abelian group. --- Absolute value. --- Addition. --- Additive group. --- Algebraic group. --- Algebraic number field. --- Algebraic number theory. --- Analytic continuation. --- Analytic function. --- Arbitrarily large. --- Automorphic form. --- Cartan subgroup. --- Class field theory. --- Complex space. --- Congruence subgroup. --- Conjugacy class. --- Coprime integers. --- Cusp form. --- Differential equation. --- Dimension (vector space). --- Direct integral. --- Direct sum. --- Division algebra. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Euler product. --- Existential quantification. --- Exponential function. --- Factorization. --- Finite field. --- Formal power series. --- Fourier series. --- Fourier transform. --- Fuchsian group. --- Function (mathematics). --- Function space. --- Functional equation. --- Fundamental unit (number theory). --- Galois extension. --- Global field. --- Group algebra. --- Group representation. --- Haar measure. --- Harish-Chandra. --- Hecke L-function. --- Hilbert space. --- Homomorphism. --- Induced representation. --- Infinite product. --- Inner automorphism. --- Integer. --- Invariant measure. --- Invariant subspace. --- Irreducible representation. --- L-function. --- Lie algebra. --- Linear map. --- Matrix coefficient. --- Mellin transform. --- Meromorphic function. --- Modular form. --- P-adic number. --- Poisson summation formula. --- Prime ideal. --- Prime number. --- Principal series representation. --- Projective representation. --- Quadratic field. --- Quadratic form. --- Quaternion algebra. --- Quaternion. --- Real number. --- Regular representation. --- Representation theory. --- Ring (mathematics). --- Ring of integers. --- Scientific notation. --- Selberg trace formula. --- Simple algebra. --- Square-integrable function. --- Sub"ient. --- Subgroup. --- Summation. --- Theorem. --- Theory. --- Theta function. --- Topological group. --- Topology. --- Trace formula. --- Trivial representation. --- Uniqueness theorem. --- Unitary operator. --- Unitary representation. --- Universal enveloping algebra. --- Upper half-plane. --- Variable (mathematics). --- Vector space. --- Weil group. --- Nombres, Théorie des

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The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. Furthermore, an explicit quantitative rate of convergence to the ergodic mean is established in many cases. The topic of this volume lies at the intersection of several mathematical fields of fundamental importance. These include ergodic theory and dynamics of non-amenable groups, harmonic analysis on semisimple algebraic groups and their homogeneous spaces, quantitative non-Euclidean lattice point counting problems and their application to number theory, as well as equidistribution and non-commutative Diophantine approximation. Many examples and applications are provided in the text, demonstrating the usefulness of the results established.

Dynamics. --- Ergodic theory. --- Harmonic analysis. --- Lattice theory. --- Lie groups. --- Ergodic theory --- Lie groups --- Lattice theory --- Harmonic analysis --- Dynamics --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Dynamical systems --- Kinetics --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Groups, Lie --- Ergodic transformations --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Banach algebras --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical --- Lie algebras --- Symmetric spaces --- Topological groups --- Continuous groups --- Mathematical physics --- Measure theory --- Absolute continuity. --- Algebraic group. --- Amenable group. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Automorphism. --- Borel set. --- Bounded function. --- Bounded operator. --- Bounded set (topological vector space). --- Congruence subgroup. --- Continuous function. --- Convergence of random variables. --- Convolution. --- Coset. --- Counting problem (complexity). --- Counting. --- Differentiable function. --- Dimension (vector space). --- Diophantine approximation. --- Direct integral. --- Direct product. --- Discrete group. --- Embedding. --- Equidistribution theorem. --- Ergodicity. --- Estimation. --- Explicit formulae (L-function). --- Family of sets. --- Haar measure. --- Hilbert space. --- Hyperbolic space. --- Induced representation. --- Infimum and supremum. --- Initial condition. --- Interpolation theorem. --- Invariance principle (linguistics). --- Invariant measure. --- Irreducible representation. --- Isometry group. --- Iwasawa group. --- Lattice (group). --- Lie algebra. --- Linear algebraic group. --- Linear space (geometry). --- Lipschitz continuity. --- Mass distribution. --- Mathematical induction. --- Maximal compact subgroup. --- Maximal ergodic theorem. --- Measure (mathematics). --- Mellin transform. --- Metric space. --- Monotonic function. --- Neighbourhood (mathematics). --- Normal subgroup. --- Number theory. --- One-parameter group. --- Operator norm. --- Orthogonal complement. --- P-adic number. --- Parametrization. --- Parity (mathematics). --- Pointwise convergence. --- Pointwise. --- Principal homogeneous space. --- Principal series representation. --- Probability measure. --- Probability space. --- Probability. --- Rate of convergence. --- Regular representation. --- Representation theory. --- Resolution of singularities. --- Sobolev space. --- Special case. --- Spectral gap. --- Spectral method. --- Spectral theory. --- Square (algebra). --- Subgroup. --- Subsequence. --- Subset. --- Symmetric space. --- Tensor algebra. --- Tensor product. --- Theorem. --- Transfer principle. --- Unit sphere. --- Unit vector. --- Unitary group. --- Unitary representation. --- Upper and lower bounds. --- Variable (mathematics). --- Vector group. --- Vector space. --- Volume form. --- Word metric.