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 series formed by the Bessel function of the first kind is known as the Schlömilch's Series.
Definition
The Fourier–Bessel series of a function f(x) with a domain of [0,b] satisfying f(b)=0
\( f:[0,b]\rightarrow {\mathbb {R}} \)
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
\( f (J_{\alpha })_{n}(x):=J_{\alpha }\left({\frac {u_{{\alpha ,n}}}b}x\right) \)
where uα,n is a root, numbered n associated with the Bessel function Jα and cn are the assigned coefficients:
\( f f(x) \sim \sum_{n=1}^\infty c_n J_\alpha \left( \frac{u_{\alpha,n}}b x \right). \)
Interpretation
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.
Calculating the coefficients
As said, differently scaled Bessel Functions are orthogonal with respect to the inner product
\( f \langle f,g\rangle =\int _{0}^{b}xf(x)g(x){\mathrm {d}}x \)
according to
\( f \int _{0}^{1}xJ_{\alpha }(xu_{{\alpha ,n}})\,J_{\alpha }(xu_{{\alpha ,m}})\,dx={\frac {\delta _{{mn}}}{2}}[J_{{\alpha +1}}(u_{{\alpha ,n}})]^{2},\)
(where: \( f {\displaystyle {\delta _{mn}}} \) is the Kronecker delta). The coefficients can be obtained from projecting the function f(x) onto the respective Bessel functions:
\( f {\displaystyle c_{n}={\frac {\langle f,(J_{\alpha })_{n}\rangle }{\langle (J_{\alpha })_{n},(J_{\alpha })_{n}\rangle }}={\frac {\int _{0}^{b}xf(x)(J_{\alpha })_{n}(x)\mathrm {d} x}{{\frac {1}{2}}(bJ_{\alpha \pm 1}(u_{\alpha ,n}))^{2}}}}\)
where the plus or minus sign is equally valid.
Application
The Fourier–Bessel series expansion employs aperiodic and decaying Bessel functions as the basis. The Fourier–Bessel series expansion has been successfully applied in diversified areas such as Gear fault diagnosis, discrimination of odorants in a turbulent ambient, postural stability analysis, detection of voice onset time, glottal closure instants (epoch) detection, separation of speech formants, EEG signal segmentation, speech enhancement, and speaker identification. The Fourier–Bessel series expansion has also been used to reduce cross terms in the Wigner–Ville distribution.
Dini series
A second Fourier–Bessel series, also known as Dini series, is associated with the Robin boundary condition
f bf'(b)+cf(b)=0, where c is an arbitrary constant.
The Dini series can be defined by
\( f f(x)\sim \sum _{{n=1}}^{\infty }b_{n}J_{\alpha }(\gamma _{n}x/b),\)
where \( f \gamma _{n} \) is the nth zero of \( f xJ'_{\alpha }(x)+cJ_{\alpha }(x). \)
The coefficients \( f b_{n} \) are given by
\( f b_n = \frac{2 \gamma_n^2}{ b^2(c^2+\gamma_n^2-\alpha^2)J_\alpha^2(\gamma_n)} \int_0^b J_\alpha(\gamma_n x/b)\,f(x) \,x\,dx. \)
See also
Orthogonality
Generalized Fourier series
Hankel transform
Neumann polynomial
References
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