# Fourier Coefficient

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## Fourier Coefficient

Fourier coefficients are the coefficients

in the Fourier series expansion of a periodic function *f*(*x*) with period 2*Ƭ* (*see*). Formulas (*) are sometimes called the Euler-Fourier formulas.

A continuous function *f*(*x*) is uniquely determined by its Fourier coefficients. The Fourier coefficients of an integrable function *f*(*x*) approach zero as *n* → ∞. Moreover, the rate of their decrease depends on the differentiability properties of *f*(*x*). For example, if *f*(*x*) has *k* continuous derivatives, then there is a number *c* such that |*a _{n}*| ≤

*cln*and |

^{k}*b*| ≤

_{n}*cln*. The Fourier coefficients are also connected with

^{k}*f*(

*x*) by the equality

(*see*PARSEVAL EQUALITY). The Fourier coefficients of a function *f*(*x*) with respect to any normalized system of functions ϕ_{1}(*x*), ϕ_{2}, . . ., ϕ* _{n}*(

*x*), . . . orthogonal on a segment [

*a*,

*b*] are given by the formula

(*see*ORTHOGONAL SYSTEM OF FUNCTIONS).