Harmonic analysis

A name given to a branch of mathematics and to a mathematical method. Harmonic analysis as a branch of mathematics is usually understood to include the theory of trigonometric series (one-dimensional and higher-dimensional); Fourier transforms (of functions of one or more variables); almost-periodic functions; Dirichlet series; approximation theory (of functions by trigonometric polynomials); abstract harmonic analysis; and certain other related mathematical disciplines (cf. Fourier transform; Almost-periodic function; Harmonic analysis, abstract). The method consists in reducing certain problems (from various fields of mathematics) to problems in harmonic analysis, which are then solved. To give an example, in the theory of functions of a complex variable problems on the boundary behaviour of functions which are analytic in the unit disc actually merge with the theory of trigonometric series; the study of the properties of random variables with the aid of characteristic functions is an application of the method of harmonic analysis to probability theory; certain objects of functional analysis are closely connected with trigonometric series, almost-periodic functions and other objects of harmonic analysis; the theory of differential equations uses the Fourier method, which belongs to harmonic analysis, to obtain solutions of various equations of mathematical physics; finally, various applied problems in numerical mathematics are solved using Fourier series and Fourier integrals (cf. Fourier integral), which are objects of harmonic analysis.

References

[a1]	J.M. Ash (ed.) , Studies in harmonic analysis , Math. Assoc. Amer. (1976)
[a2]	Y. Katznelson, "An introduction to harmonic analysis" , Dover, reprint (1976)