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Singularities of solutions of differential equations forms the common theme of these papers taken from a seminar held at the Institute for Advanced Study in Princeton in 1977-1978. While some of the lectures were devoted to the analysis of singularities, others focused on applications in spectral theory. As an introduction to the subject, this volume treats current research in the field in such a way that it can be studied with profit by the non-specialist.

Partial differential equations --- Differential equations, Linear --- Differential equations, Partial --- Equations différentielles linéaires --- Equations aux dérivées partielles --- Numerical solutions --- Congresses --- Solutions numériques --- Congrès --- Théorie asymptotique --- 517.95 --- -Differential equations, Partial --- -Partial differential equations --- Linear differential equations --- Linear systems --- Insect societies. --- Insects --- Congresses. --- Ecology. --- 517.95 Partial differential equations --- -517.95 Partial differential equations --- Equations différentielles linéaires --- Equations aux dérivées partielles --- Solutions numériques --- Congrès --- Théorie asymptotique --- -Hexapoda --- Insecta --- Pterygota --- Arthropoda --- Entomology --- Behavior, Animal --- Ecology --- Insecta. --- Insect societies --- Sociétés d'insectes --- Insectes --- Ecologie --- Numerical solutions&delete& --- Insects, Social --- Social insects --- Animal societies --- Behavior --- Insects. Springtails --- Animal ethology and ecology. Sociobiology --- Behavior, Animal. --- Équations aux dérivées partielles --- Solutions numériques. --- A priori estimate. --- Adjoint equation. --- Analytic continuation. --- Analytic function. --- Analytic manifold. --- Asymptote. --- Asymptotic analysis. --- Asymptotic distribution. --- Asymptotic expansion. --- Asymptotic formula. --- Big O notation. --- Calculus on manifolds. --- Canonical transformation. --- Characteristic equation. --- Characteristic function (probability theory). --- Codimension. --- Cohomology. --- Commutator. --- Complex manifold. --- Complex number. --- Continuous function (set theory). --- Continuous function. --- Covariant derivative. --- Diffeomorphism. --- Differential equation. --- Differential operator. --- Dirichlet problem. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elementary proof. --- Elliptic boundary value problem. --- Equation. --- Equivalence class. --- Equivalence relation. --- Error term. --- Existence theorem. --- Existential quantification. --- Exponential function. --- Fourier integral operator. --- Fourier inversion theorem. --- Fourier transform. --- Functional calculus. --- Fundamental solution. --- Hamiltonian vector field. --- Hardy space. --- Harmonic analysis. --- Hermann Weyl. --- Hermitian adjoint. --- Hilbert space. --- Holomorphic function. --- Homogeneous function. --- Hyperbolic partial differential equation. --- Hyperfunction. --- Hypersurface. --- Inclusion map. --- Inequality (mathematics). --- Integer lattice. --- Integral transform. --- Irreducible representation. --- Lagrangian (field theory). --- Laplace operator. --- Limit (mathematics). --- Linear map. --- Local diffeomorphism. --- Manifold. --- Mathematical optimization. --- Maximal torus. --- Monotonic function. --- Ordinary differential equation. --- Oscillatory integral. --- Partial differential equation. --- Partition of unity. --- Poisson bracket. --- Poisson summation formula. --- Polynomial. --- Projection (linear algebra). --- Projective variety. --- Pseudo-differential operator. --- Regularity theorem. --- Renormalization. --- Riemann surface. --- Riemannian manifold. --- Riesz representation theorem. --- Self-adjoint operator. --- Self-adjoint. --- Sign (mathematics). --- Special case. --- Spectral theorem. --- Spectral theory. --- Summation. --- Support (mathematics). --- Symplectic geometry. --- Symplectic manifold. --- Taylor series. --- Theorem. --- Toeplitz operator. --- Trace class. --- Trigonometric polynomial. --- Unit disk. --- Variable (mathematics). --- Equations aux derivees partielles lineaires

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The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.

Harmonic analysis. --- Harmonic functions. --- Functions, Harmonic --- Laplace's equations --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Harmonic analysis. Fourier analysis --- Harmonic analysis --- Fourier analysis --- Harmonic functions --- Analyse harmonique --- Analyse de Fourier --- Fonctions harmoniques --- Fourier Analysis --- Fourier, Transformations de --- Euclide, Espaces d' --- Bessel functions --- Differential equations, Partial --- Fourier series --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Banach algebras --- Time-series analysis --- Analysis, Fourier --- Fourier analysis. --- Basic Sciences. Mathematics --- Analysis, Functions --- Analysis, Functions. --- Calculus --- Mathematical analysis --- Mathematics --- Fourier, Transformations de. --- Euclide, Espaces d'. --- Potentiel, Théorie du --- Fonctions harmoniques. --- Potential theory (Mathematics) --- Analytic continuation. --- Analytic function. --- Banach algebra. --- Banach space. --- Bessel function. --- Borel measure. --- Boundary value problem. --- Bounded operator. --- Bounded set (topological vector space). --- Cartesian coordinate system. --- Cauchy–Riemann equations. --- Change of variables. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Complex plane. --- Conformal map. --- Conjugate transpose. --- Continuous function (set theory). --- Continuous function. --- Convolution. --- Differentiation of integrals. --- Dimensional analysis. --- Dirichlet problem. --- Disk (mathematics). --- Distribution (mathematics). --- Equation. --- Euclidean space. --- Existential quantification. --- Fourier inversion theorem. --- Fourier series. --- Fourier transform. --- Fubini's theorem. --- Function (mathematics). --- Function space. --- Green's theorem. --- Hardy's inequality. --- Hardy–Littlewood maximal function. --- Harmonic function. --- Hermitian matrix. --- Hilbert transform. --- Holomorphic function. --- Homogeneous function. --- Inequality (mathematics). --- Infimum and supremum. --- Interpolation theorem. --- Interval (mathematics). --- Lebesgue integration. --- Lebesgue measure. --- Linear interpolation. --- Linear map. --- Linear space (geometry). --- Line–line intersection. --- Liouville's theorem (Hamiltonian). --- Lipschitz continuity. --- Locally integrable function. --- Lp space. --- Majorization. --- Marcinkiewicz interpolation theorem. --- Mean value theorem. --- Measure (mathematics). --- Mellin transform. --- Monotonic function. --- Multiplication operator. --- Norm (mathematics). --- Operator norm. --- Orthogonal group. --- Paley–Wiener theorem. --- Partial derivative. --- Partial differential equation. --- Plancherel theorem. --- Pointwise convergence. --- Poisson kernel. --- Poisson summation formula. --- Polynomial. --- Principal value. --- Quadratic form. --- Radial function. --- Radon–Nikodym theorem. --- Representation theorem. --- Riesz transform. --- Scientific notation. --- Series expansion. --- Singular integral. --- Special case. --- Subharmonic function. --- Support (mathematics). --- Theorem. --- Topology. --- Total variation. --- Trigonometric polynomial. --- Trigonometric series. --- Two-dimensional space. --- Union (set theory). --- Unit disk. --- Unit sphere. --- Upper half-plane. --- Variable (mathematics). --- Vector space. --- Fourier, Analyse de --- Potentiel, Théorie du. --- Potentiel, Théorie du --- Espaces de hardy