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Fourier Integrals in Classical Analysis is an advanced monograph concerned with modern treatments of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. Using microlocal analysis, the author, in particular, studies problems involving maximal functions and Riesz means using the so-called half-wave operator. This self-contained book starts with a rapid review of important topics in Fourier analysis. The author then presents the necessary tools from microlocal analysis, and goes on to give a proof of the sharp Weyl formula which he then modifies to give sharp estimates for the size of eigenfunctions on compact manifolds. Finally, at the end, the tools that have been developed are used to study the regularity properties of Fourier integral operators, culminating in the proof of local smoothing estimates and their applications to singular maximal theorems in two and more dimensions.

Fourier series. --- Fourier integral operators.

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This advanced monograph is concerned with modern treatments of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. In particular, the author uses microlocal analysis to study problems involving maximal functions and Riesz means using the so-called half-wave operator. To keep the treatment self-contained, the author begins with a rapid review of Fourier analysis and also develops the necessary tools from microlocal analysis. This second edition includes two new chapters. The first presents Hörmander's propagation of singularities theorem and uses this to prove the Duistermaat-Guillemin theorem. The second concerns newer results related to the Kakeya conjecture, including the maximal Kakeya estimates obtained by Bourgain and Wolff.

Fourier series. --- Fourier integral operators. --- Fourier analysis. --- Analysis, Fourier --- Mathematical analysis --- Fourier analysis --- Integral operators --- Fourier integrals --- Series, Fourier --- Series, Trigonometric --- Trigonometric series --- Calculus --- Harmonic analysis --- Harmonic functions

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Complex analysis --- Harmonic analysis. Fourier analysis --- Fourier integral operators --- Mathematical analysis. --- Opérateurs intégraux de Fourier --- Analyse mathématique --- Fourier integral operators. --- 51 <082.1> --- Mathematics--Series --- Opérateurs intégraux de Fourier --- Analyse mathématique --- Mathematical analysis --- 517.1 Mathematical analysis --- Fourier analysis --- Integral operators --- 517.1. --- 517.1

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Fourier integral operators --- Opérateurs intégraux de Fourier --- Fourier integral operators. --- Opérateurs intégraux de Fourier --- Differential equations, Partial --- Équations aux dérivées partielles --- Pseudodifferential operators --- Opérateurs pseudo-différentiels --- Équations aux dérivées partielles. --- Opérateurs pseudo-différentiels. --- Fourier, Opérateurs intégraux de --- Équations aux dérivées partielles

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Partial differential equations -- Equations of mathematical physics and other areas of application -- PDEs in connection with quantum mechanics --- Quantum theory -- General mathematical topics and methods in quantum theory -- Semiclassical techniques, including WKB and Maslov methods --- Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Pseudodifferential operators --- Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Fourier integral operators --- Partial differential equations -- Spectral theory and eigenvalue problems -- Asymptotic distribution of eigenvalues and eigenfunctions --- Quantum theory -- General quantum mechanics and problems of quantization -- Geometry and quantization, symplectic methods --- Quantum theory --- Differential equations, Partial --- Théorie quantique --- Equations aux dérivées partielles --- Mathematics --- Mathématiques --- Differential equations, Partial. --- Partial differential equations -- Equations of mathematical physics and other areas of application -- PDEs in connection with quantum mechanics. --- Quantum theory -- General mathematical topics and methods in quantum theory -- Semiclassical techniques, including WKB and Maslov methods. --- Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Pseudodifferential operators. --- Partial differential equations -- Pseudodifferential operators and other generalizations of partial differential operators -- Fourier integral operators. --- Partial differential equations -- Spectral theory and eigenvalue problems -- Asymptotic distribution of eigenvalues and eigenfunctions. --- Quantum theory -- General quantum mechanics and problems of quantization -- Geometry and quantization, symplectic methods. --- Mathematics. --- Théorie quantique --- Equations aux dérivées partielles --- Mathématiques --- Manifolds (Mathematics) --- Variétés (mathématiques) --- Équations aux dérivées partielles. --- Pseudodifferential operators --- Opérateurs pseudo-différentiels --- Pseudodifferential operators. --- Opérateurs pseudo-différentiels --- Variétés (mathématiques)