43: Abstract harmonic analysis
If Fourier series is the study of periodic real functions , that is, real functions whichare 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.
Subfields

• 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.
• 43A20 : L^1-algebras on groups, semigroups, etc.

 
46: Functional analysis
Functional analysis views the big picture in differential equations, for example, thinking of a differential operator as a linear map on a large set of functions. Thus this area becomes the study of (infinite-dimensional) vector spaces with some kind of metric or other structure, including ring structures (Banach algebras and C-* algebras for example). Appropriate generalizations of measure, derivatives, and duality also belong to this area.
Subfields

• 46G: Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces), see also 28-XX For nonlinear functional analysis, See 47HXX, 58-XX, especially 58CXX
• 46H: Topological algebras, normed rings and algebras, Banach algebras, For group algebras, convolution algebras and measure algebras, see 43A10, 43A20
• 46J: Commutative Banach algebras and commutative topological algebras, see also 46E25

 
The Other Research Interests:
42: Fourier analysis
47: Operator theory