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HARMONIC ANALYSIS

DOI: 10.1615/AtoZ.h.harmonic_analysis

Harmonic analysis is a branch of mathematics, which includes theories of trigonometric series (Fourier Series), Fourier transformations, function approximation by trigonometric polynomials, almost periodic functions, and also generalization of these notions in connection with general problems of the theory of functions and functional analysis.

Each periodic function f(t) having a period T and satisfying Dirichlet's conditions (a discontinuity of the first kind, a finite and a countable number of extremums in a period) can be represented (expanded) in the form of a sum of an infinite number of sinusoidal functions

(1)

where the frequency ω = 2π/T, Ai and ai are constant.

The components of expansion (1) are called the harmonic components (harmonics of the lth, 2nd etc. kind), and the expansion itself, the harmonic analysis of the function f(t).

If we denote x = ωt, then expansion (1) for the function f(x) with a period 2π has the form

and after transformations

(2)

which is called the Fourier series.

The coefficients in (2) are defined as

Series (2) converges to f(x) at points of its continuity, and to a half-sum [f(xi ) + f(xi+)]/2 at discontinuity points xi.

The expansion of an even function (f(–x) = f(x)) into a Fourier series has only cosines, and of an odd function (f(–x) = –f(x)), only sines.

In general, the expansion of a function f(x) into a series according to any orthogonal system of functions {φn(x)}, defined on the interval [a,b],

with coefficients defined as

is called the generalized Fourier series of a given function, and the coefficients cn the generalized Fourier coefficients relative to expansion {φn(x)}.

For an arbitrary orthogonal on the interval [a,b] system of square integrable functions {φn(x)}, the least average deviation for a given m

of the function f from an approximating polynomial

is obtained, if as σm is used a truncated series of a generalized Fourier series

Then

If {φn} = {sin nx; cos nx}, then a minimum Δm is obtained, if the Fourier truncated series is taken and

Similarly, each function f(x) defined on a number axis (–∞,∞) which satisfies the Dirichlet conditions can be represented in the form of the Fourier integral

with the right-hand part being equal at discontinuity points to [f(x) + f(x+)]/2.

The Fourier integral is also called the Fourier transform of function F(s) which is the inverse Fourier transform of f(x).

A Fourier series and a Fourier integral relate each function f(x) and its spectrum (for a Fourier integral a set of harmonics is the spectrum). In some approaches (for instance, in turbulence theory) the study of functions by their spectra is used.

The notion of Fourier transformation can be easily generalized into the functions of many variables.

Harmonic analysis methods and similar methods associated with the expansion in terms of a complete orthogonal system of functions, are widely used in solving the problems of mathematical physics, associated with the solution of parabolic and elliptic partial differential equations.

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