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Concept
Fourier–Bessel series
Résumé
In mathematics, Fourier–Bessel series is a particular kind of generalized Fourier series (an infinite series expansion on a finite interval) based on Bessel functions.
Fourier–Bessel series are used in the solution to partial differential equations, particularly in cylindrical coordinate systems.
The Fourier–Bessel series of a function f(x) with a domain of satisfying f(b) = 0
is the representation of that function as a linear combination of many orthogonal versions of the same Bessel function of the first kind Jα, where the argument to each version n is differently scaled, according to
where uα,n is a root, numbered n associated with the Bessel function Jα and cn are the assigned coefficients:
The Fourier–Bessel series may be thought of as a Fourier expansion in the ρ coordinate of cylindrical coordinates. Just as the Fourier series is defined for a finite interval and has a counterpart, the continuous Fourier transform over an infinite interval, so the Fourier–Bessel series has a counterpart over an infinite interval, namely the Hankel transform.
As said, differently scaled Bessel Functions are orthogonal with respect to the inner product
according to
(where: is the Kronecker delta). The coefficients can be obtained from projecting the function f(x) onto the respective Bessel functions:
where the plus or minus sign is equally valid.
For the inverse transform, one makes use of the following representation of the Dirac delta function
Fourier–Bessel series coefficients are unique for a given signal, and there is one-to-one mapping between continuous frequency () and order index which can be expressed as follows:
Since, . So above equation can be rewritten as follows:
where is the length of the signal and is the sampling frequency of the signal.
Source officielle
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