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Symplectic Methods in Harmonic Analysis and in Mathematical Physics
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Harmonic analysis is one of the most active and fastest growing parts of both pure and applied mathematics. It has gone far beyond its primary goal, which was to study the representation of functions or signals as superpositions of trigonometric functions (Fourier series). The interest in harmonic analysis has always been great because of the wealth of its applications, and it plays nowadays a central role in the study of signal theory and time-frequency analysis. Its interest in pure mathematics (especially in functional analysis) has been revived by the introduction of new functional spaces which are tools of choice for studying regularity properties of pseudo-differential operators and their applications to mathematical physics. Methods from symplectic geometry add power and scope to modern harmonic analysis; historically these methods were perhaps for the first time systematically used in Folland’s seminal book [59].
The aim of the present book is to give a rigorous and modern treatment of various objects from harmonic analysis with a strong emphasis on the underlying symplectic structure (for instance symplectic and metaplectic covariance proper- ties). More specifically we have in mind two audiences: the time-frequency community, and mathematical physicists interested in applications to quantum mechanics. The concepts and methods are presented in such a way that they should be easily accessible to students at the upper-undergraduate level (a certain familiarity with basic Fourier analysis and the elementary theory of distributions is assumed). Needless to say, this book can also be read with profit by more advanced readers, and can be used as a reference work by researchers in partial differential equations, harmonic analysis, and mathematical physics. (Several chapters are part of ongoing research and contain material that is usually not addressed in introductory texts. For instance Gromov’s non-squeezing theorem from symplectic topology and its applications, or the theory of phase space pseudo differential operators.)
Maurice A. de Gosson - Personal Name
1st Edtion
978-3-7643-9992-4
NONE
Symplectic Methods in Harmonic Analysis and in Mathematical Physics
Mathematics
English
Springer Basel
2011
1-362
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