nLab harmonic analysis

Idea

The basic problem of harmonic analysis is the decomposition of elements in some topological vector space of functions in some linear basis which is typically distinguished by some nice representation theoretical properties. This decomposition can be a sum, and a basis a topological basis, but more general it is a decomposition in the sense of an integral. The elements of the distinguished bases were in historical examples thought of as “basic waves” or “harmonics”. Some standard examples are Fourier analysis on locally compact abelian groups, wavelet analysis?, quantum group Fourier transform etc. In some cases the elements of the “basis” are not linearly independent, e.g. in the case of decomposition into coherent states.

Related entries

• Fourier transform, Fourier-Laplace transform

• Fourier transform of distributions

• Fourier analysis,

• Bochner's theorem

• Tannaka-Krein theorem,

• Peter-Weyl theorem,

• Gelfand-Raikov theorem,

• sampling theory

• Restriction Theory

References

• Edwin Hewett, Kenneth Ross, Abstract Harmonic Analysis

  Volume I: Structure of Topological Groups, Integration Theory, Group Representations, Grundlehren der mathematischen Wissenschaften 115, Springer 1979 (doi:10.1007/978-1-4419-8638-2)

  Volume II: Structure and Analysis for Compact Groups, Analysis on Locally Compact Abelian Groups, Grundlehren der mathematischen Wissenschaften 152, Springer 1970 (doi:10.1007/978-3-662-26755-4)

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press (1995) [pdf, gBooks]

• Elias Stein, Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press 1971

See also:

• Wikipedia, Harmonic analysis

• MSRI, Categorical Structures in Harmonic Analysis, Nov 2014. (videos)

category: analysis