欧几里得空间的傅立叶分析
2009-8
世界图书出版公司
stein
297
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This book is designed to be an introduction to harmonic analysis inEuclidean spaces. The subject has seen a considerable flowering during thepast twenty years. We have not tried to cover all phases of this develop-ment. Rather, our chief concern was to illustrate various methods used inthis aspect of Fourier analysis that exploit the structure of Euclideanspaces. In particular, we try to show the role played by the action oftranslations, dilations, and rotations. Another concern, not independentof this chief one, is to motivate the study of harmonic analysis on moregeneral spaces having an analogous structure （such as arises in symmetricspaces）. It is our feeling that the study of Fourier analysis in that contextand, also, in other general settings, is more meaningful once the specialEuclidean case is understood.
PrefaceCHAPTER Ⅰ The Fourier Transform 1. The basic L1 theory of the Fourier transform 2.The L2 theory and the Plancherel theorem 3.The class of tempered distributions 4.Further resultsCHAPTER Ⅱ Boundary Values of Harmonic Functions 1.Basic properties of harmonic functions 2.The characterization of Poisson integrals 3.The Hardy-Littlewood maximal function and nontangential convergence of harmonic functions 4.Subharmonic functions and majorization by harmonic functions 5.Further resultsCHAPTER Ⅲ The Theory of Hp Spaces on Tubes 1.Introductory remarks 2.The H2 theory 3.Tubes over cones 4.The Paley-Wiener theorem 5.The Hp theory 6.Further resultsCHAPTER Ⅳ.Symmetry Properties of the Fourier Transform 1.Decomposition of L2(Ez) into'sub, paces invariant under the Fourier transform 2.Spherical harmonics 3.The action of the Fourier transform on the spaces 4.Some applications 5.Further resultsCHAPTER Ⅴ Interpolation of Operators 1.The M. Riesz convexity theorem and interpolation of operators defined on Lp spaces 2.The Marcinkiewicz interpolation theorem 3.L(p, q) spaces 4.Interpolation of analytic families of operators 5.Further resultsCHAPTER Ⅵ Singular Integrals and Systems of Conjugate Harmonic Functions 1.The Hilbert transform 2.Singular integral operators with odd kernels 3.Singular integral operators with even kernels 4.Hp spaces of conjugate harmonic functions 5.Further resultsCHAPTER Ⅶ Multiple Fourier Series 1.Elementary properties 2.The Poisson summation formula 3.Multiplier transformations 4.Summability below the critical index (negative results) 5.Summability below the critical index 6.Further resultsBibliographyIndex
We require that the readers have mastered the material that is usuallycovered in a course in the theory of integration and in the theory offunctions of a complex variable.
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