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# Theory of Functions and related areas

Here we mean the study of vector spaces of functions. This can include the abstract study of topological vector spaces as well as the study of particular spaces of interest, including attention to their bases (e.g. Fourier Analysis), and linear maps on them (e.g. Integral Transforms).

• 46: 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.
• 42: Fourier analysis studies approximations and decompositions of functions using trigonometric polynomials. Of incalculable value in many applications of analysis, this field has grown to include many specific and powerful results, including convergence criteria, estimates and inequalities, and existence and uniqueness results. Extensions include the theory of singular integrals, Fourier transforms, and the study of the appropriate function spaces. This heading also includes approximations by other orthogonal families of functions, including orthogonal polynomials and wavelets.
• 43: 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 groups which are invariant under a 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.
• 44: Integral transforms include the Fourier transform (see above) as well as the transforms of Laplace, Radon, and others. (The general theory of transformations between function spaces is part of Functional Analysis, above.) Also includes convolution operators and operational calculi.
• 47: Operator theory studies transformations between the vector spaces studied in Functional Analysis, such as differential operators or self-adjoint operators. The analysis might study the spectrum of an individual operator or the semigroup structure of a collection of them.

Also highlighted on the MathMap is 22: Topological Groups since, in practice, a good deal of material in that area is concerned with Harmonic Analysis -- the study of special (bases of) functions on Lie Groups.

Clearly related to this topic are several of the fields of classical analysis and calculus, such as 39: Functional Equations and 33: Special Functions. We have (somewhat arbitrarily) grouped the fields in this area together because of the emphasis on the vector-space perspective.

You might want to continue the tour with a trip through differential equations.