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FOURIER SERIES

DOI: 10.1615/AtoZ.f.fourier_series

The Fourier series of the function f in terms of a system of functions {φn(t)} orthonormal on the interval [a, b] is an infinite series

(1a)

whose coefficients are determined by

(1)

and are called the Fourier coefficients of the function f. The function f is in the general case supposed to be integrable over the interval [a, b]. More often than not, a system of trigonometric functions used as the orthonormal system of functions. Thus, a Fourier series in a trigonometric system is usually what we mean by Fourier's series.

If an interval of expansion is given − π < t < π, then the Fourier series for a real function f(t), is an infinite trigonometric series

(2)

whose coefficients are defined by

(3)
(4)

Here ak, bk are real, and are, generally speaking, complex numbers. The Fourier coefficients ak, bk of the function f(t) for k → ∞ tend to zero.

In case of expansion of an even function (f(t) = f(–t)) into a Fourier series, it has only cosines, and of odd function (f(t) = –f(–t)) it has only sines.

If a trigonometric series (2) converges to f(t) uniformly, then the coefficients are necesserily the Fourier coefficients (3) of f(t).

Among the trigonometric polynomials of order n

(5)

the least value for the root-mean-square error

(6)

is reached when as Tn(t) a partial sum of a Fourier series, for the function f

(7)

a system of trigonometric functions is complete (closed) and Parseval's equation holds for it

(8)

If a function f(t) on a finite interval satisfies the Dirichlet's conditions (i.e., has a finite number of extremums and is continuous everywhere, besides the finite number of points in which in can have discontinuities of the first kind), then Fourier series of function f converges for all t, during which it converges to f(t) at continuity points, and to a half-sum 1/2 [f(t – 0) + f(t + 0)] at discontinuity points.

This statement is extended into arbitrary function of bounded variation (i.e., into functions representable in the form f = f1 – f2, where f1 and f2 are bounded and decreases on [a, b]). If the function f(x) is continuous and has a bounded variation, then a Fourier series converges to this function uniformly (the Dirichlet-Jordan test).

A specific property of Fourier series, associated with peculiarities in its behavior (an excess of partial sums of Fourier series over the exact value of the function) in the vicinity of discontinuity points of the function f(t) is called Gibb's phenomenon. Gibb's phenomenon is associated with nonuniform convergence, limits the application of Fourier series in the vicinity of discontinuity points and requires additional effort in order to ensure reliable approximate computations in this field (the summation over arithmetic means).

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