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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.

## Subfields

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

• 43A05: Measures on groups and semigroups, etc.
• 43A07: Means on groups, semigroups, etc.; amenable groups
• 43A10: Measure algebras on groups, semigroups, etc.
• 43A15: L^p-spaces and other function spaces on groups, semigroups, etc.
• 43A17: Analysis on ordered groups, H^p-theory
• 43A20: L^1-algebras on groups, semigroups, etc.
• 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
• 43A25: Fourier and Fourier-Stieltjes transforms on locally compact abelian groups
• 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
• 43A32: Other transforms and operators of Fourier type
• 43A35: Positive definite functions on groups, semigroups, etc.
• 43A40: Character groups and dual objects
• 43A45: Spectral synthesis on groups, semigroups, etc.
• 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
• 43A50: Convergence of Fourier series and of inverse transforms
• 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
• 43A62: Hypergroups
• 43A65: Representations of groups, semigroups, etc., See Also 22A10, 22A20, 22Dxx, 22E45
• 43A70: Analysis on specific locally compact abelian groups, See also 11R56, 22B05
• 43A75: Analysis on specific compact groups
• 43A77: Analysis on general compact groups
• 43A85: Analysis on homogeneous spaces
• 43A99: Miscellaneous topics

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