How to calculate the Fourier transform of the Kaiser-Bessel window? According to Wikipedia: https://en.wikipedia.org/wiki/Kaiser_window, the Fourier transform of the Kaiser-Bessel window
$w_0(x) :=  \left\{ \begin{array}{**lr**} \frac{I_0(\pi \alpha \sqrt{1-{(2x/L)}^2})}{I_0(\pi \alpha)}, &  |x| \leq \frac{L}{2}\\  
             0, & |x| > \frac{L}{2}\\  
             \end{array} \right.  $
is
$W(f) := \frac{L\cdot \sinh(\pi\alpha\sqrt{1-(Lf/\alpha)^2})}{I_0(\pi\alpha) \cdot \pi\alpha\sqrt{1-(Lf/\alpha)^2}}$,
where $I_0(x) := \sum\limits_{m=0}^{\infty}\frac{1}{(m!)^2}(\frac{x}{2})^{2m}$ is the zeroth-order modified Bessel function of the first kind (https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_I%CE%B1,_K%CE%B1).
I have made many attempts but fail to deduce the closed-form solution of the Fourier transform of the Kaiser-Bessel window. First, I tried to expand $I_0(x)$, which was quite difficult to calculate since it involved infinite series. Second,  I tried to calculate the Fourier transform from the opposite side, i.e., $W(f)$, but got no result. Finally, I tried to use Mathematica Professional 12, but it failed to offer a result after long time calculation.
I have also found some literature online, e.g.,
[1] On the use of the I0-sinh window for spectrum analysis, IEEE Transactions on Acoustics, Speech, and Signal Processing, J. Kaiser, R. Schafer, 1980, doi: 10.1109/TASSP.1980.1163349.
[2] On the use of windows for harmonic analysis with the discrete Fourier transform, Proceedings of the IEEE, F.J. Harris, 1978, doi: 10.1109/PROC.1978.10837.
[3] https://www.dsprelated.com/freebooks/sasp/Kaiser_Window.html
[4] Verification of Fourier transformation of Io-sinh function (MathStackExchange; No answer yet).
Neither of them deduces the closed form of the Fourier transform of the Kaiser-Bessel window. Some references, e.g., [1] and [3], point to an old unavailable book named "System Analysis by Digital Computer". Kaiser-Bessel window is a widely used window function in signal processing. However, it is really difficult to find relevant literature online. Additionally, I am not sure whether the Fourier transform provided in  https://en.wikipedia.org/wiki/Kaiser_window is a closed-form or an approximation. I deeply appreciate your help or hints.
 A: Using a parity property, the Fourier integral can be written as
\begin{equation}
 K=\frac{2}{I_0(\pi\alpha)}\int_0^{L/2}I_0\left(\pi \alpha \sqrt{1-{(2x/L)}^2}\right)\cos(2\pi fx)\,dx
\end{equation}
We use the quoted series expansion for the modified Bessel function to obtain after swaping integration and summation
\begin{align}
 K&=\frac{2}{I_0(\pi\alpha)}\sum_{m=0}^\infty \frac{1}{(m!)^2}\left( \frac{\pi \alpha}{2} \right)^{2m}\int_0^{L/2}\left( 1-(2x/L)^2 \right)^{m}\cos(2\pi fx)\,dx\\
 &=\frac{L}{I_0(\pi\alpha)}\sum_{m=0}^\infty \frac{1}{(m!)^2}\left( \frac{\pi \alpha}{2} \right)^{2m}\int_0^1\left( 1-t^2 \right)^{m}\cos(t\pi fL)\,dt
\end{align}
This cosine transform is tabulated in Ederlyi (TI 1.3.8) or can be related to an integral representation of the Bessel function:
\begin{equation}
 J_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}%
\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}%
\cos\left(zt\right)\mathrm{d}t
\end{equation}
With $\nu=m+1/2,z=\pi fL$ one obtains
\begin{equation}
 \int_0^1\left( 1-t^2 \right)^{m}\cos(t\pi fL)\,dt=m!\sqrt{\pi}2^{-m+1/2}(\pi fL)^{-m-1/2}J_{m+1/2}(\pi f L)
\end{equation}
Then, after some simplifications,
\begin{equation}
 K=\frac{L}{I_0(\pi\alpha)\sqrt{2fL}}\sum_{m=0}^\infty \frac{1}{m!}\left( \pi \alpha\right)^{2m}2^{-m}(\pi fL)^{-m}J_{m+1/2}(\pi f L)
\end{equation}
Such a series looks similar to the multiplication theorem for the Bessel functions:
\begin{equation}
J_{\nu}\left(\lambda Z\right)=\lambda^{\nu}\sum_{m=0}^{\infty}%
\frac{(- 1)^{m}(\lambda^{2}-1)^{m}(\tfrac{1}{2}Z)^{m}}{m!}J_{\nu+ m}\left(Z\right)
\end{equation}
which is valid for any complex value of $\lambda$. We use $\nu=1/2,Z=\pi fL$ and $\lambda=\sqrt{1-\frac{\alpha^2}{f^2L^2}}$ with $\Im\lambda\ge0$ to write
\begin{equation}
 K=\frac{L}{I_0(\pi\alpha)\sqrt{2fL}}\frac{1}{\left( 1-\frac{\alpha^2}{f^2L^2} \right)^{1/4}}J_{1/2}\left( \pi fL\sqrt{1-\frac{\alpha^2}{f^2L^2}} \right)
\end{equation}
and with the explicit expression for $J_{1/2}$,
\begin{equation}
  K=\frac{L}{I_0(\pi\alpha)}\frac{\sin\left( \pi\sqrt{f^2L^2-\alpha^2} \right)}{\pi\sqrt{f^2L^2-\alpha^2} }
\end{equation}
which is valid for all the values of $f$. In particular, for $f<\alpha/L$, it is convenient to write the above expression as
\begin{equation}
   K=\frac{L}{I_0(\pi\alpha)}\frac{\sinh\left( \pi\alpha\sqrt{1-f^2L^2/\alpha^2} \right)}{\pi\alpha\sqrt{1-f^2L^2/\alpha^2} }
\end{equation}
which is the proposed expression for the Fourier transform.
