Fourier–Bessel series

The Fourier–Bessel series of a function f(x) with a domain of [0,b] satisfying f(b)=0

${\displaystyle f:[0,b]\rightarrow \mathbb {R} }$

is the representation of that function as a linear combination of many orthogonal versions of the same Bessel function of the first kind J_α, where the argument to each version n is differently scaled, according to

${\displaystyle (J_{\alpha })_{n}(x):=J_{\alpha }\left({\frac {u_{\alpha ,n}}{b}}x\right)}$

where u_α,n is a root, numbered n associated with the Bessel function J_α and c_n are the assigned coefficients:

${\displaystyle f(x)\sim \sum _{n=1}^{\infty }c_{n}J_{\alpha }\left({\frac {u_{\alpha ,n}}{b}}x\right).}$

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

${\displaystyle \langle f,g\rangle =\int _{0}^{b}xf(x)g(x)\mathrm {d} x}$

according to

${\displaystyle \int _{0}^{1}xJ_{\alpha }(xu_{\alpha ,n})\,J_{\alpha }(xu_{\alpha ,m})\,dx={\frac {\delta _{mn}}{2}}[J_{\alpha +1}(u_{\alpha ,n})]^{2}}$,

(where: ${\displaystyle {\delta _{mn}}}$ is the Kronecker delta). The coefficients can be obtained from projecting the function f(x) onto the respective Bessel functions:

${\displaystyle c_{n}={\frac {\langle f,(J_{\alpha })_{n}\rangle }{\langle (J_{\alpha })_{n},(J_{\alpha })_{n}\rangle }}={\frac {\int _{0}^{b}xf(x)(J_{\alpha })_{n}(x)\mathrm {d} x}{{\frac {1}{2}}(bJ_{\alpha \pm 1}(u_{\alpha ,n}))^{2}}}}$

where the plus or minus sign is equally valid.

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.

A second Fourier–Bessel series, also known as Dini series, is associated with the Robin boundary condition

${\displaystyle bf'(b)+cf(b)=0}$, where ${\displaystyle c}$ is an arbitrary constant.

The Dini series can be defined by

${\displaystyle f(x)\sim \sum _{n=1}^{\infty }b_{n}J_{\alpha }(\gamma _{n}x/b)}$,

where ${\displaystyle \gamma _{n}}$ is the nth zero of ${\displaystyle xJ'_{\alpha }(x)+cJ_{\alpha }(x)}$.

The coefficients ${\displaystyle b_{n}}$ are given by

${\displaystyle b_{n}={\frac {2\gamma _{n}^{2}}{b^{2}(c^{2}+\gamma _{n}^{2}-\alpha ^{2})J_{\alpha }^{2}(\gamma _{n})}}\int _{0}^{b}J_{\alpha }(\gamma _{n}x/b)\,f(x)\,x\,dx.}$

• Orthogonality
• Generalized Fourier series
• Hankel transform
• Neumann polynomial

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