Fourier Coefficient

Fourier coefficients are the coefficients

in the Fourier series expansion of a periodic function f(x) with period 2Ƭ (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^k and |b_n| ≤ cln^k. The Fourier coefficients are also connected with f(x) by the equality

(seePARSEVAL EQUALITY). The Fourier coefficients of a function f(x) with respect to any normalized system of functions ϕ₁(x), ϕ₂, . . ., ϕ_n(x), . . . orthogonal on a segment [a, b] are given by the formula

(seeORTHOGONAL SYSTEM OF FUNCTIONS).