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* Updated material on Hausdorff dimension and fractal dimension.
Mathematical analysis --- Functions of real variables --- 517.51 --- Advanced calculus --- Analysis (Mathematics) --- Algebra --- Real variables --- Functions of complex variables --- Functions of a real variable. Real functions --- Functions of real variables. --- Mathematical analysis. --- 517.1 Mathematical analysis --- 517.51 Functions of a real variable. Real functions --- 517.1. --- 517.1
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This book provides the first coherent account of the area of analysis that involves the Heisenberg group, quantization, the Weyl calculus, the metaplectic representation, wave packets, and related concepts. This circle of ideas comes principally from mathematical physics, partial differential equations, and Fourier analysis, and it illuminates all these subjects. The principal features of the book are as follows: a thorough treatment of the representations of the Heisenberg group, their associated integral transforms, and the metaplectic representation; an exposition of the Weyl calculus of pseudodifferential operators, with emphasis on ideas coming from harmonic analysis and physics; a discussion of wave packet transforms and their applications; and a new development of Howe's theory of the oscillator semigroup.
Harmonic analysis. Fourier analysis --- Phase space (Statistical physics) --- Harmonic analysis --- 512.54 <043> --- 530.145 <043> --- 517.986.6 --- 51-7 <043> --- 517.986.6 <043> --- Groups. Group theory--Dissertaties --- Quantum theory--Dissertaties --- Harmonic analysis of functions of groups and homogeneous spaces --- Mathematical studies and methods in other sciences. Scientific mathematics. Actuarial mathematics. Biometrics. Econometrics etc.--Dissertaties --- Harmonic analysis of functions of groups and homogeneous spaces--Dissertaties --- 517.986.6 <043> Harmonic analysis of functions of groups and homogeneous spaces--Dissertaties --- 51-7 <043> Mathematical studies and methods in other sciences. Scientific mathematics. Actuarial mathematics. Biometrics. Econometrics etc.--Dissertaties --- 517.986.6 Harmonic analysis of functions of groups and homogeneous spaces --- 530.145 <043> Quantum theory--Dissertaties --- 512.54 <043> Groups. Group theory--Dissertaties --- Space, Phase (Statistical physics) --- Generalized spaces --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Harmonic analysis. --- Analytic continuation. --- Analytic function. --- Antisymmetric tensor. --- Asymptotic expansion. --- Automorphism. --- Bilinear form. --- Bounded operator. --- Calculation. --- Canonical commutation relation. --- Canonical transformation. --- Cauchy–Riemann equations. --- Cayley transform. --- Class function (algebra). --- Classical mechanics. --- Commutative property. --- Complex analysis. --- Configuration space. --- Differential equation. --- Differential geometry. --- Differential operator. --- Eigenvalues and eigenvectors. --- Equation. --- Explicit formula. --- Fock space. --- Fourier analysis. --- Fourier integral operator. --- Fourier transform. --- Functional analysis. --- Gaussian function. --- Gaussian integral. --- Geometric quantization. --- Hamiltonian mechanics. --- Hamiltonian vector field. --- Heisenberg group. --- Hermite polynomials. --- Hermitian symmetric space. --- Hilbert space. --- Hilbert transform. --- Integral transform. --- Invariant subspace. --- Irreducible representation. --- Lebesgue measure. --- Lie algebra. --- Lie superalgebra. --- Lie theory. --- Mathematical physics. --- Number theory. --- Observable. --- Ordinary differential equation. --- Orthonormal basis. --- Oscillator representation. --- Oscillatory integral. --- Partial differential equation. --- Phase factor. --- Phase space. --- Point at infinity. --- Poisson bracket. --- Polynomial. --- Power series. --- Probability. --- Projection (linear algebra). --- Projective Hilbert space. --- Projective representation. --- Projective space. --- Pseudo-differential operator. --- Pullback (category theory). --- Quadratic function. --- Quantum harmonic oscillator. --- Quantum mechanics. --- Representation theory. --- Schrödinger equation. --- Self-adjoint operator. --- Semigroup. --- Several complex variables. --- Siegel disc. --- Sobolev space. --- Spectral theorem. --- Spectral theory. --- State-space representation. --- Stone's theorem. --- Stone–Weierstrass theorem. --- Summation. --- Symmetric space. --- Symmetric tensor. --- Symplectic geometry. --- Symplectic group. --- Symplectic vector space. --- Symplectomorphism. --- Tangent space. --- Tangent vector. --- Theorem. --- Translational symmetry. --- Unbounded operator. --- Unit vector. --- Unitarity (physics). --- Unitary operator. --- Unitary representation. --- Variable (mathematics). --- Wave packet.
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An in-depth look at real analysis and its applications-now expanded and revised.This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory.This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include:* Revised material on the n-dimensional Lebesgue integral.* An improved proof of Tychonoff's theorem.* Expanded material on Fourier analysis.* A newly written chapter devoted to distributions and differential equations.* Updated material on Hausdorff dimension and fractal dimension.
Analyse (Mathématique) --- Analyse (Wiskunde) --- Analyse mathématique --- Analysis (Mathematics) --- Fonctions de variables reelles --- Functies van reële veranderlijken --- Functions of real variables --- Mathematical analysis --- Wiskundige analyse --- Functions of real variables. --- Mathematical analysis.
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A Course in Abstract Harmonic Analysis is an introduction to that part of analysis on locally compact groups that can be done with minimal assumptions on the nature of the group. As a generalization of classical Fourier analysis, this abstract theory creates a foundation for a great deal of modern analysis, and it contains a number of elegant results and techniques that are of interest in their own right.This book develops the abstract theory along with a well-chosen selection of concrete examples that exemplify the results and show the breadth of their applicability. After a preliminary chapter containing the necessary background material on Banach algebras and spectral theory, the text sets out the general theory of locally compact groups and their unitary representations, followed by a development of the more specific theory of analysis on Abelian groups and compact groups. There is an extensive chapter on the theory of induced representations and its applications, and the book concludes with a more informal exposition on the theory of representations of non-Abelian, non-compact groups.
Harmonic analysis --- Mathematical analysis --- 517.1 Mathematical analysis --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis
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The second edition of Introduction to Partial Differential Equations, which originally appeared in the Princeton series Mathematical Notes, serves as a text for mathematics students at the intermediate graduate level. The goal is to acquaint readers with the fundamental classical results of partial differential equations and to guide them into some aspects of the modern theory to the point where they will be equipped to read advanced treatises and research papers. This book includes many more exercises than the first edition, offers a new chapter on pseudodifferential operators, and contains additional material throughout. The first five chapters of the book deal with classical theory: first-order equations, local existence theorems, and an extensive discussion of the fundamental differential equations of mathematical physics. The techniques of modern analysis, such as distributions and Hilbert spaces, are used wherever appropriate to illuminate these long-studied topics. The last three chapters introduce the modern theory: Sobolev spaces, elliptic boundary value problems, and pseudodifferential operators.
Differential equations, Partial. --- Bessel function. --- Cauchy data. --- Dirichlet problem. --- Feynman propagator. --- Gaussian kernel. --- Hadamard example. --- Laplacian. --- Lewy equation. --- Neumann problem. --- Poisson kernel. --- Rellich's theorem. --- Sobolev norm. --- Sobolev space. --- adjoint. --- amplitude. --- cotangent space. --- elliptic operator. --- heat operator. --- maximum principle. --- normal derivative. --- oriented hypersurface. --- parametrix. --- radial function. --- wave operator. --- weak solution.
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