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
43A55: Summability methods on groups, semigroups, etc., See Also 40J05
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
43A80: Analysis on other specific Lie groups, See also 22Exx
43A85: Analysis on homogeneous spaces
43A90: Spherical functions, See also 22E45, 22E46, 33C65, 33D55
43A95: Categorical methods, See also 46Mxx
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
Software and tables
Other web sites with this focus
Here are the AMS and Goettingen resource pages for area 43.
Selected Topics at this site
Plancherel's theorem: the Fourier transform is an isometry.
Computing the volume element on GL_n(R).
Invariant (Haar) measures on SO(3) and SE(3) -- some summaries
Haar measure and rotation group SO(n)
SNAG (Stone-Naimark-Ambrose-Godement) Theorem: construct measures corresponding to representations of LCA groups
