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 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 |an| ≤ clnk and |bn| ≤ clnk. 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, . . ., ϕn(x), . . . orthogonal on a segment [a, b] are given by the formula

(seeORTHOGONAL SYSTEM OF FUNCTIONS).

References in periodicals archive ?
The following theorem gives the distribution of the Fourier coefficients and their relations under the general Levy-based model.
Now since [DELTA]P is symmetric, the level-2 Fourier coefficient of [DELTA]P equals,
There is an effective correlation between the Fourier coefficients [[DELTA].
In order to use the Fourier coefficients as features, frequencies at which the corresponding coefficients do not overlap are chosen as features.
The dimension of the zeros padded Fourier coefficient matrix [[?
The thus obtained Fourier coefficients of the structurally broadened profiles were scaled such that the first Fourier coefficient (correlation distance zero) was equal to one.
In view of these estimates, with T = 5 we build a 256 x 256 matrix A corresponding to truncation of the WSS sample reproducing matrix, and a corresponding 256 x 256 system producing PSWF samples as Fourier coefficients of 256 degree Legendre approximations of the PSWFs.
out] output amplitude of the fundamental component can be calculated as a function of input amplitude by evaluating the first Fourier coefficient of the expansion.
Meanwhile, according to LPTV theory, we can have the below relationship of nth order frequency-dependent Fourier coefficient [G.
In other words, the Fourier coefficient of any plane wave in the Fourier series of f [member of] B([OMEGA]) on [-L, L] with frequency > [OMEGA] is given by the above formula.
Furthermore, assume that the Fourier coefficients {u(w, [DELTA]t)} of u(x, [DELTA]t) = exp[-L[DELTA]t] f (x) satisfy an estimate