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In this book the authors give the first necessary and sufficient conditions for the uniform convergence a.s. of random Fourier series on locally compact Abelian groups and on compact non-Abelian groups. They also obtain many related results. For example, whenever a random Fourier series converges uniformly a.s. it also satisfies the central limit theorem. The methods developed are used to study some questions in harmonic analysis that are not intrinsically random. For example, a new characterization of Sidon sets is derived.The major results depend heavily on the Dudley-Fernique necessary and sufficient condition for the continuity of stationary Gaussian processes and on recent work on sums of independent Banach space valued random variables. It is noteworthy that the proofs for the Abelian case immediately extend to the non-Abelian case once the proper definition of random Fourier series is made. In doing this the authors obtain new results on sums of independent random matrices with elements in a Banach space. The final chapter of the book suggests several directions for further research.

Harmonic analysis. Fourier analysis --- Fourier series. --- Harmonic analysis. --- Fourier, Séries de --- Analyse harmonique --- 517.518.4 --- Fourier series --- Harmonic analysis --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Harmonic functions --- Time-series analysis --- Fourier integrals --- Series, Fourier --- Series, Trigonometric --- Trigonometric series --- Fourier analysis --- 517.518.4 Trigonometric series --- Fourier, Séries de --- Abelian group. --- Almost periodic function. --- Almost surely. --- Banach space. --- Big O notation. --- Cardinality. --- Central limit theorem. --- Circle group. --- Coefficient. --- Commutative property. --- Compact group. --- Compact space. --- Complex number. --- Continuous function. --- Corollary. --- Discrete group. --- Equivalence class. --- Existential quantification. --- Finite group. --- Gaussian process. --- Haar measure. --- Independence (probability theory). --- Inequality (mathematics). --- Integer. --- Irreducible representation. --- Non-abelian group. --- Non-abelian. --- Normal distribution. --- Orthogonal group. --- Orthogonal matrix. --- Probability distribution. --- Probability measure. --- Probability space. --- Probability. --- Random function. --- Random matrix. --- Random variable. --- Rate of convergence. --- Real number. --- Ring (mathematics). --- Scientific notation. --- Set (mathematics). --- Slepian's lemma. --- Small number. --- Smoothness. --- Stationary process. --- Subgroup. --- Subset. --- Summation. --- Theorem. --- Uniform convergence. --- Unitary matrix. --- Variance.

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This book is a sequel to Lectures on Complex Analytic Varieties: The Local Paranwtrization Theorem (Mathematical Notes 10, 1970). Its unifying theme is the study of local properties of finite analytic mappings between complex analytic varieties; these mappings are those in several dimensions that most closely resemble general complex analytic mappings in one complex dimension. The purpose of this volume is rather to clarify some algebraic aspects of the local study of complex analytic varieties than merely to examine finite analytic mappings for their own sake.Originally published in 1970.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.

Complex analysis --- Analytic spaces --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Spaces, Analytic --- Analytic functions --- Functions of several complex variables --- Algebra homomorphism. --- Algebraic curve. --- Algebraic extension. --- Algebraic surface. --- Algebraic variety. --- Analytic continuation. --- Analytic function. --- Associated prime. --- Atlas (topology). --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Branch point. --- Change of variables. --- Characterization (mathematics). --- Codimension. --- Coefficient. --- Cohomology. --- Complete intersection. --- Complex analysis. --- Complex conjugate. --- Complex dimension. --- Complex number. --- Connected component (graph theory). --- Corollary. --- Critical point (mathematics). --- Diagram (category theory). --- Dimension (vector space). --- Dimension. --- Disjoint union. --- Divisor. --- Equation. --- Equivalence class. --- Exact sequence. --- Existential quantification. --- Finitely generated module. --- Geometry. --- Hamiltonian mechanics. --- Holomorphic function. --- Homeomorphism. --- Homological dimension. --- Homomorphism. --- Hypersurface. --- Ideal (ring theory). --- Identity element. --- Induced homomorphism. --- Inequality (mathematics). --- Injective function. --- Integral domain. --- Invertible matrix. --- Irreducible component. --- Isolated singularity. --- Isomorphism class. --- Jacobian matrix and determinant. --- Linear map. --- Linear subspace. --- Local ring. --- Mathematical induction. --- Mathematics. --- Maximal element. --- Maximal ideal. --- Meromorphic function. --- Modular arithmetic. --- Module (mathematics). --- Module homomorphism. --- Monic polynomial. --- Monomial. --- Neighbourhood (mathematics). --- Noetherian. --- Open set. --- Parametric equation. --- Parametrization. --- Permutation. --- Polynomial ring. --- Polynomial. --- Power series. --- Quadratic form. --- Quotient module. --- Regular local ring. --- Removable singularity. --- Ring (mathematics). --- Ring homomorphism. --- Row and column vectors. --- Scalar multiplication. --- Scientific notation. --- Several complex variables. --- Sheaf (mathematics). --- Special case. --- Subalgebra. --- Submanifold. --- Subset. --- Summation. --- Surjective function. --- Taylor series. --- Theorem. --- Three-dimensional space (mathematics). --- Topological space. --- Vector space. --- Weierstrass preparation theorem. --- Zero divisor. --- Fonctions de plusieurs variables complexes --- Variétés complexes

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In 1970, Phillip Griffiths envisioned that points at infinity could be added to the classifying space D of polarized Hodge structures. In this book, Kazuya Kato and Sampei Usui realize this dream by creating a logarithmic Hodge theory. They use the logarithmic structures begun by Fontaine-Illusie to revive nilpotent orbits as a logarithmic Hodge structure. The book focuses on two principal topics. First, Kato and Usui construct the fine moduli space of polarized logarithmic Hodge structures with additional structures. Even for a Hermitian symmetric domain D, the present theory is a refinement of the toroidal compactifications by Mumford et al. For general D, fine moduli spaces may have slits caused by Griffiths transversality at the boundary and be no longer locally compact. Second, Kato and Usui construct eight enlargements of D and describe their relations by a fundamental diagram, where four of these enlargements live in the Hodge theoretic area and the other four live in the algebra-group theoretic area. These two areas are connected by a continuous map given by the SL(2)-orbit theorem of Cattani-Kaplan-Schmid. This diagram is used for the construction in the first topic.

Hodge theory. --- Logarithms. --- Logs (Logarithms) --- Algebra --- Complex manifolds --- Differentiable manifolds --- Geometry, Algebraic --- Homology theory --- Algebraic group. --- Algebraic variety. --- Analytic manifold. --- Analytic space. --- Annulus (mathematics). --- Arithmetic group. --- Atlas (topology). --- Canonical map. --- Classifying space. --- Coefficient. --- Cohomology. --- Compactification (mathematics). --- Complex manifold. --- Complex number. --- Congruence subgroup. --- Conjecture. --- Connected component (graph theory). --- Continuous function. --- Convex cone. --- Degeneracy (mathematics). --- Diagram (category theory). --- Differential form. --- Direct image functor. --- Divisor. --- Elliptic curve. --- Equivalence class. --- Existential quantification. --- Finite set. --- Functor. --- Geometry. --- Hodge structure. --- Homeomorphism. --- Homomorphism. --- Inverse function. --- Iwasawa decomposition. --- Local homeomorphism. --- Local ring. --- Local system. --- Logarithmic. --- Maximal compact subgroup. --- Modular curve. --- Modular form. --- Moduli space. --- Monodromy. --- Monoid. --- Morphism. --- Natural number. --- Nilpotent orbit. --- Nilpotent. --- Open problem. --- Open set. --- P-adic Hodge theory. --- P-adic number. --- Point at infinity. --- Proper morphism. --- Pullback (category theory). --- Quotient space (topology). --- Rational number. --- Relative interior. --- Ring (mathematics). --- Ring homomorphism. --- Scientific notation. --- Set (mathematics). --- Sheaf (mathematics). --- Smooth morphism. --- Special case. --- Strong topology. --- Subgroup. --- Subobject. --- Subset. --- Surjective function. --- Tangent bundle. --- Taylor series. --- Theorem. --- Topological space. --- Topology. --- Transversality (mathematics). --- Two-dimensional space. --- Vector bundle. --- Vector space. --- Weak topology.

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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

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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