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43: Abstract harmonic analysis


Introduction

Abstract harmonic analysis: if Fourier series is the study of periodic real functions, that is, real functions which are invariant under the group of integer translations, then abstract harmonic analysis is the study of functions on general topological groups which are invariant under a (closed) subgroup. This includes topics of varying level of specificity, including analysis on Lie groups or locally compact abelian groups. This area also overlaps with representation theory of topological groups.

History

Mackey, George W. : "Harmonic analysis as the exploitation of symmetry---a historical survey", Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 1, part 1, 543--698 (MR81d:01017)

Applications and related fields

For other analysis on topological and Lie groups, See 22Exx

One can carry over the development of Fourier series for functions on the circle and study the expansion of functions on the sphere; the basic functions then are the spherical harmonics -- see 33: Special Functions. [Schematic of subareas and related areas]

Subfields

There is only one division (43A) but it is subdivided:

This is among the smaller areas in the Math Reviews database.

Browse all (old) classifications for this area at the AMS.


Textbooks, reference works, and tutorials

Berenstein, Carlos A.: "The Pompeiu problem, what's new?", Complex analysis, harmonic analysis and applications (Bordeaux, 1995), 1--11; Pitman Res. Notes Math. Ser., 347; Longman, Harlow, 1996. MR97g:43007

Software and tables

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