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Harmonic analysis. Fourier analysis --- Mathematical physics --- Fourier analysis --- Analyse de Fourier --- Fourier analysis. --- Fourier Analysis. --- Fourier transformations --- Fourier series --- Functions, Special --- Laplace transformation --- Fourier, Transformations de --- Fonctions spéciales --- Transformation de Laplace --- Fourier, Séries de --- Fourier, Analyse de --- Fourier, Transformations de. --- Fonctions spéciales. --- Transformation de Laplace. --- Fourier, Séries de. --- Fourier, Analyse de. --- Fourier Analysis

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Quantum mechanics. Quantumfield theory --- Quantum electrodynamics --- Quantum field theory --- Théorie quantique des champs --- Mathematics. --- Mathématiques --- Mathematics --- Théorie quantique des champs --- Mathématiques --- Quantum electrodynamics - Mathematics --- Quantum field theory - Mathematics

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A Guide to Advanced Real Analysis is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form.

Mathematical analysis. --- Functions of real variables. --- Real variables --- Functions of complex variables --- 517.1 Mathematical analysis --- Mathematical analysis --- Functions of real variables

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Part explanation of important recent work, and part introduction to some of the techniques of modern partial differential equations, this monograph is a self-contained exposition of the Neumann problem for the Cauchy-Riemann complex and certain of its applications. The authors prove the main existence and regularity theorems in detail, assuming only a knowledge of the basic theory of differentiable manifolds and operators on Hilbert space. They discuss applications to the theory of several complex variables, examine the associated complex on the boundary, and outline other techniques relevant to these problems. In an appendix they develop the functional analysis of differential operators in terms of Sobolev spaces, to the extent it is required for the monograph.

Functional analysis --- Neumann problem --- Differential operators --- Complex manifolds --- Complex manifolds. --- Differential operators. --- Neumann problem. --- Differential equations, Partial --- Équations aux dérivées partielles --- Analytic spaces --- Manifolds (Mathematics) --- Operators, Differential --- Differential equations --- Operator theory --- Boundary value problems --- A priori estimate. --- Almost complex manifold. --- Analytic function. --- Apply. --- Approximation. --- Bernhard Riemann. --- Boundary value problem. --- Calculation. --- Cauchy–Riemann equations. --- Cohomology. --- Compact space. --- Complex analysis. --- Complex manifold. --- Coordinate system. --- Corollary. --- Derivative. --- Differentiable manifold. --- Differential equation. --- Differential form. --- Differential operator. --- Dimension (vector space). --- Dirichlet boundary condition. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Equation. --- Estimation. --- Euclidean space. --- Existence theorem. --- Exterior (topology). --- Finite difference. --- Fourier analysis. --- Fourier transform. --- Frobenius theorem (differential topology). --- Functional analysis. --- Hilbert space. --- Hodge theory. --- Holomorphic function. --- Holomorphic vector bundle. --- Irreducible representation. --- Line segment. --- Linear programming. --- Local coordinates. --- Lp space. --- Manifold. --- Monograph. --- Multi-index notation. --- Nonlinear system. --- Operator (physics). --- Overdetermined system. --- Partial differential equation. --- Partition of unity. --- Potential theory. --- Power series. --- Pseudo-differential operator. --- Pseudoconvexity. --- Pseudogroup. --- Pullback. --- Regularity theorem. --- Remainder. --- Scientific notation. --- Several complex variables. --- Sheaf (mathematics). --- Smoothness. --- Sobolev space. --- Special case. --- Statistical significance. --- Sturm–Liouville theory. --- Submanifold. --- Tangent bundle. --- Theorem. --- Uniform norm. --- Vector field. --- Weight function. --- Operators in hilbert space

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The object of this monograph is to give an exposition of the real-variable theory of Hardy spaces (HP spaces). This theory has attracted considerable attention in recent years because it led to a better understanding in Rn of such related topics as singular integrals, multiplier operators, maximal functions, and real-variable methods generally. Because of its fruitful development, a systematic exposition of some of the main parts of the theory is now desirable. In addition to this exposition, these notes contain a recasting of the theory in the more general setting where the underlying Rn is replaced by a homogeneous group.The justification for this wider scope comes from two sources: 1) the theory of semi-simple Lie groups and symmetric spaces, where such homogeneous groups arise naturally as "boundaries," and 2) certain classes of non-elliptic differential equations (in particular those connected with several complex variables), where the model cases occur on homogeneous groups. The example which has been most widely studied in recent years is that of the Heisenberg group.

Functions of real variables. --- Hardy spaces. --- Lie groups. --- "admissible. --- Campanato space. --- Campbell-Hausdorff formul. --- Chebyshev's inequality. --- Constants. --- Derivatives and multiindices. --- Dilations. --- Hardy space. --- Hardy-Littlewood maximal function. --- Heisenberg group. --- Littlewood-Paley function. --- Lusin function. --- Maximal functions. --- Norms. --- Other operations on functions". --- Poisson kernel. --- Poisson-Szegö kernel. --- area integral. --- associated (to a ball). --- atom. --- atomic Hardy space. --- atomic decomposition. --- ball. --- commutative approximate identity. --- convolution. --- dilations. --- distribution function. --- graded. --- grand maximal function. --- heat kernel. --- heat semigroup. --- isotropic degree. --- kernel of type. --- lower central series. --- nilpotent. --- nonincreasing rearrangement. --- nontangential maximal function. --- p-admissible. --- polynomial. --- polyradial. --- positive operator. --- quasinorms. --- seminorms.