Fourier–Bessel series

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

$${\displaystyle f:[0,b]\to \mathbb {R} }$$ 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 ^[1][2] $${\displaystyle (J_{\alpha })_{n}(x):=J_{\alpha }\left({\frac {u_{\alpha ,n}}{b}}x\right)}$$ where u_α,n is a root, numbered n associated with the Bessel function J_α and c_n are the assigned coefficients:^[3] $${\displaystyle f(x)\sim \sum _{n=1}^{\infty }c_{n}J_{\alpha }\left({\frac {u_{\alpha ,n}}{b}}x\right).}$$

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.

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As said, differently scaled Bessel Functions are orthogonal with respect to the inner product

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

according to

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

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

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

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^[4]

$${\displaystyle {\frac {2x^{\alpha }y^{1-\alpha }}{b^{2}}}\sum _{k=1}^{\infty }{\frac {J_{\alpha }\left({\frac {xu_{\alpha ,k}}{b}}\right)\,J_{\alpha }\left({\frac {yu_{\alpha ,k}}{b}}\right)}{J_{\alpha +1}^{2}(u_{\alpha ,k})}}=\delta (x-y).}$$

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,^[5] discrimination of odorants in a turbulent ambient,^[6] postural stability analysis, detection of voice onset time, glottal closure instants (epoch) detection, separation of speech formants, speech enhancement,^[7] and speaker identification.^[8] The Fourier–Bessel series expansion has also been used to reduce cross terms in the Wigner–Ville distribution.

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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 n-th 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
• Kapteyn series
• Neumann polynomial
• Schlömilch's series