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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 presents a development of the basic facts about harmonic analysis on local fields and the n-dimensional vector spaces over these fields. It focuses almost exclusively on the analogy between the local field and Euclidean cases, with respect to the form of statements, the manner of proof, and the variety of applications.The force of the analogy between the local field and Euclidean cases rests in the relationship of the field structures that underlie the respective cases. A complete classification of locally compact, non-discrete fields gives us two examples of connected fields (real and complex numbers); the rest are local fields (p-adic numbers, p-series fields, and their algebraic extensions). The local fields are studied in an effort to extend knowledge of the reals and complexes as locally compact fields.The author's central aim has been to present the basic facts of Fourier analysis on local fields in an accessible form and in the same spirit as in Zygmund's Trigonometric Series (Cambridge, 1968) and in Introduction to Fourier Analysis on Euclidean Spaces by Stein and Weiss (1971).Originally published in 1975.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.

Fourier analysis. --- Local fields (Algebra) --- Fields, Local (Algebra) --- Algebraic fields --- Analysis, Fourier --- Mathematical analysis --- Corps algébriques --- Fourier analysis --- 511 --- 511 Number theory --- Number theory --- Local fields (Algebra). --- Harmonic analysis. Fourier analysis --- Fourier Analysis --- Abelian group. --- Absolute continuity. --- Absolute value. --- Addition. --- Additive group. --- Algebraic extension. --- Algebraic number field. --- Bessel function. --- Beta function. --- Borel measure. --- Bounded function. --- Bounded variation. --- Boundedness. --- Calculation. --- Cauchy–Riemann equations. --- Characteristic function (probability theory). --- Complex analysis. --- Conformal map. --- Continuous function. --- Convolution. --- Coprime integers. --- Corollary. --- Coset. --- Determinant. --- Dimension (vector space). --- Dimension. --- Dirichlet kernel. --- Discrete space. --- Distribution (mathematics). --- Endomorphism. --- Field of fractions. --- Finite field. --- Formal power series. --- Fourier series. --- Fourier transform. --- Gamma function. --- Gelfand. --- Haar measure. --- Haar wavelet. --- Half-space (geometry). --- Hankel transform. --- Hardy's inequality. --- Harmonic analysis. --- Harmonic function. --- Homogeneous distribution. --- Integer. --- Lebesgue integration. --- Linear combination. --- Linear difference equation. --- Linear map. --- Linear space (geometry). --- Local field. --- Lp space. --- Maximal ideal. --- Measurable function. --- Measure (mathematics). --- Mellin transform. --- Metric space. --- Modular form. --- Multiplicative group. --- Norbert Wiener. --- P-adic number. --- Poisson kernel. --- Power series. --- Prime ideal. --- Probability. --- Product metric. --- Rational number. --- Regularization (mathematics). --- Requirement. --- Ring (mathematics). --- Ring of integers. --- Scalar multiplication. --- Scientific notation. --- Sign (mathematics). --- Smoothness. --- Special case. --- Special functions. --- Subgroup. --- Subring. --- Support (mathematics). --- Theorem. --- Topological space. --- Unitary operator. --- Vector space. --- Analyse harmonique (mathématiques) --- Analyse harmonique (mathématiques) --- Corps algébriques