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Functional analysis --- 51 <082.1> --- Mathematics--Series --- Operator algebras. --- Operator spaces. --- Operator ideals. --- Algèbres d'opérateurs --- Espaces d'opérateurs --- Idéaux d'opérateurs --- Operator algebras --- Operator ideals --- Operator spaces --- Matricially normed spaces --- Non-commutative Banach space theory --- Banach spaces --- Operator theory --- Ideals (Algebra) --- Algebras, Operator --- Topological algebras --- Algèbres d'opérateurs. --- Espaces d'opérateurs. --- Idéaux d'opérateurs.

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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 contains an exposition of some of the main developments of the last twenty years in the following areas of harmonic analysis: singular integral and pseudo-differential operators, the theory of Hardy spaces, Lsup estimates involving oscillatory integrals and Fourier integral operators, relations of curvature to maximal inequalities, and connections with analysis on the Heisenberg group.

Harmonic analysis. Fourier analysis --- Harmonic analysis --- Analyse harmonique --- Harmonic analysis. --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Groupe de Heisenberg. --- Addition. --- Analytic function. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Asymptotic formula. --- Automorphism. --- Axiom. --- Banach space. --- Bessel function. --- Big O notation. --- Bilinear form. --- Borel measure. --- Boundary value problem. --- Bounded function. --- Bounded mean oscillation. --- Bounded operator. --- Boundedness. --- Cancellation property. --- Cauchy's integral theorem. --- Cauchy–Riemann equations. --- Characteristic polynomial. --- Characterization (mathematics). --- Commutative property. --- Commutator. --- Complex analysis. --- Convolution. --- Differential equation. --- Differential operator. --- Dimension (vector space). --- Dimension. --- Dirac delta function. --- Dirichlet problem. --- Elliptic operator. --- Existential quantification. --- Fatou's theorem. --- Fourier analysis. --- Fourier integral operator. --- Fourier inversion theorem. --- Fourier series. --- Fourier transform. --- Fubini's theorem. --- Function (mathematics). --- Fundamental solution. --- Gaussian curvature. --- Hardy space. --- Harmonic function. --- Heisenberg group. --- Hilbert space. --- Hilbert transform. --- Holomorphic function. --- Hölder's inequality. --- Infimum and supremum. --- Integral transform. --- Interpolation theorem. --- Lagrangian (field theory). --- Laplace's equation. --- Lebesgue measure. --- Lie algebra. --- Line segment. --- Linear map. --- Lipschitz continuity. --- Locally integrable function. --- Marcinkiewicz interpolation theorem. --- Martingale (probability theory). --- Mathematical induction. --- Maximal function. --- Meromorphic function. --- Multiplication operator. --- Nilpotent Lie algebra. --- Norm (mathematics). --- Number theory. --- Operator theory. --- Order of integration (calculus). --- Orthogonality. --- Oscillatory integral. --- Poisson summation formula. --- Projection (linear algebra). --- Pseudo-differential operator. --- Pseudoconvexity. --- Rectangle. --- Riesz transform. --- Several complex variables. --- Sign (mathematics). --- Singular integral. --- Sobolev space. --- Special case. --- Spectral theory. --- Square (algebra). --- Stochastic differential equation. --- Subharmonic function. --- Submanifold. --- Summation. --- Support (mathematics). --- Theorem. --- Translational symmetry. --- Uniqueness theorem. --- Variable (mathematics). --- Vector field. --- Fourier, Analyse de --- Fourier, Opérateurs intégraux de

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Written by Arthur Ogus on the basis of notes from Pierre Berthelot's seminar on crystalline cohomology at Princeton University in the spring of 1974, this book constitutes an informal introduction to a significant branch of algebraic geometry. Specifically, it provides the basic tools used in the study of crystalline cohomology of algebraic varieties in positive characteristic.Originally published in 1978.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.

Algebraic geometry --- Geometry, Algebraic. --- Homology theory. --- Functions, Zeta. --- Zeta functions --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Geometry --- Abelian category. --- Additive map. --- Adjoint functors. --- Adjunction (field theory). --- Adjunction formula. --- Alexander Grothendieck. --- Algebra homomorphism. --- Artinian. --- Automorphism. --- Axiom. --- Banach space. --- Base change map. --- Base change. --- Betti number. --- Calculation. --- Cartesian product. --- Category of abelian groups. --- Characteristic polynomial. --- Characterization (mathematics). --- Closed immersion. --- Codimension. --- Coefficient. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Commutative property. --- Commutative ring. --- Compact space. --- Corollary. --- Crystalline cohomology. --- De Rham cohomology. --- Degeneracy (mathematics). --- Derived category. --- Diagram (category theory). --- Differential operator. --- Discrete valuation ring. --- Divisibility rule. --- Dual basis. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Epimorphism. --- Equation. --- Equivalence of categories. --- Exact sequence. --- Existential quantification. --- Explicit formula. --- Explicit formulae (L-function). --- Exponential type. --- Exterior algebra. --- Exterior derivative. --- Formal power series. --- Formal scheme. --- Frobenius endomorphism. --- Functor. --- Fundamental theorem. --- Hasse invariant. --- Hodge theory. --- Homotopy. --- Ideal (ring theory). --- Initial and terminal objects. --- Inverse image functor. --- Inverse limit. --- Inverse system. --- K-theory. --- Leray spectral sequence. --- Linear map. --- Linearization. --- Locally constant function. --- Mapping cone (homological algebra). --- Mathematical induction. --- Maximal ideal. --- Module (mathematics). --- Monomial. --- Monotonic function. --- Morphism. --- Natural transformation. --- Newton polygon. --- Noetherian ring. --- Noetherian. --- P-adic number. --- Polynomial. --- Power series. --- Presheaf (category theory). --- Projective module. --- Scientific notation. --- Series (mathematics). --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Spectral sequence. --- Subring. --- Subset. --- Symmetric algebra. --- Theorem. --- Topological space. --- Topology. --- Topos. --- Transitive relation. --- Universal property. --- Zariski topology. --- Geometrie algebrique --- Topologie algebrique --- Varietes algebriques --- Cohomologie

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The present volume reflects both the diversity of Bochner's pursuits in pure mathematics and the influence his example and thought have had upon contemporary researchers.Originally published in 1971.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 analysis --- -Advanced calculus --- Analysis (Mathematics) --- Algebra --- Addresses, essays, lectures --- Mathematical analysis. --- -517.1 Mathematical analysis --- -Addresses, essays, lectures --- 517.1 Mathematical analysis --- 517.1. --- Approximation theory. --- System analysis. --- 517.1 --- Network analysis --- Network science --- Network theory --- Systems analysis --- System theory --- Mathematical optimization --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Absolute continuity. --- Analytic continuation. --- Analytic function. --- Asymptotic expansion. --- Automorphism. --- Banach algebra. --- Banach space. --- Bessel function. --- Big O notation. --- Bounded operator. --- Branch point. --- Cauchy's integral formula. --- Cauchy's integral theorem. --- Characterization (mathematics). --- Cohomology. --- Commutative property. --- Compact operator. --- Compact space. --- Complex analysis. --- Complex number. --- Complex plane. --- Continuous function (set theory). --- Continuous function. --- Convolution. --- Coset. --- Covariance operator. --- Differentiable function. --- Differentiable manifold. --- Differential form. --- Dimension (vector space). --- Discrete group. --- Dominated convergence theorem. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Equation. --- Equivalence class. --- Even and odd functions. --- Existential quantification. --- First variation. --- Formal power series. --- Fréchet derivative. --- Fubini's theorem. --- Function space. --- Functional analysis. --- Fundamental group. --- Green's function. --- Haar measure. --- Hermite polynomials. --- Hermitian symmetric space. --- Holomorphic function. --- Hyperbolic partial differential equation. --- Infimum and supremum. --- Infinite product. --- Integral equation. --- Lebesgue measure. --- Lie algebra. --- Lie group. --- Linear map. --- Markov chain. --- Meromorphic function. --- Metric space. --- Monotonic function. --- Natural number. --- Norm (mathematics). --- Normal subgroup. --- Null set. --- Partition of unity. --- Pointwise. --- Polynomial. --- Power series. --- Pseudogroup. --- Riemann surface. --- Riemannian manifold. --- Schrödinger equation. --- Self-adjoint operator. --- Self-adjoint. --- Semigroup. --- Semisimple algebra. --- Sesquilinear form. --- Sign (mathematics). --- Singular perturbation. --- Special case. --- Spectral theory. --- Stokes' theorem. --- Subgroup. --- Submanifold. --- Subset. --- Support (mathematics). --- Symplectic geometry. --- Symplectic manifold. --- Theorem. --- Uniform convergence. --- Unitary operator. --- Unitary representation. --- Upper and lower bounds. --- Vector bundle. --- Vector field. --- Volterra's function. --- Weierstrass theorem. --- Zorn's lemma.