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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.