harmonic analysis

The basic problem of **harmonic analysis** is the decomposition of elements in some topological vector space of functions in some 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.

- $n$Lab: Fourier analysis, Tannaka-Krein theorem, Peter-Weyl theorem?, Gelfand-Raikov theorem, sampling theory
- Wikipedia: harmonic analysis
- Gerald B. Folland,
*A course in abstract harmonic analysis*, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. x+276 pp. gBooks - Elias Stein, Guido Weiss,
*Introduction to Fourier analysis on Euclidean spaces*, Princeton Univ. Press 1971

category: analysis

Revised on May 16, 2013 20:00:00
by Zoran Škoda
(161.53.130.104)