In this paper, an oscillatory radial basis function is introduced and its properties are considered. The RBF is as follows: $$\phi_d=\frac{J_{d/2-1}(\epsilon r)}{(\epsilon r)^{d/2-1}},\quad d=2,3,4,...$$ in which $J_\nu$ denotes the $J$-Bessel function of order $\nu$ and $\epsilon$ is a constant, called a shape parameter. The authors took the Fourier transform of $\phi_d$ which is as follows: $$\hat{\phi}_d(\parallel \frac{\overline{\xi}}{\epsilon}\parallel_2)=\int_\mathbb{R^n}\phi_d(\epsilon\parallel\overline{x}\parallel_2)e^{-i(\overline{x}.\overline{\xi})}d\overline{x}$$ $$ =\frac{(1-\parallel\overline{\xi}\parallel_{2}^{2}/\epsilon^2)^{((d-n)/2-1)}}{\epsilon^n2^{(d/2-1)}\Gamma((d-n)/2)\pi^{n/2}},\quad if \parallel \frac{\overline{\xi}}{\epsilon}\parallel_2\leq1$$ and for $\parallel \frac{\overline{\xi}}{\epsilon}\parallel_2\gt1$ the result is $0$ in which $\overline{x}=[x_1,...,x_n]$ and $\overline{\xi}=[\xi_1,...,\xi_n]$ and $\parallel.\parallel$ is the standard Euclidean vector norm.
Now my question is how the integral above is calculated? I tried Fourier transform of the radially symmetric functions, related to Hankel transform, using this note but I arrived to: $$\hat{\phi}_d(s)=\frac{(2\pi)^{n/2}}{s^{n/2-1}}\int\limits_0^\infty J_{n/2-1}(sr)r^{n/2-1}\phi_d rdr$$ which I think is related to hyperspherical coordinates when place the $\phi_d$ in the integral, but I can't continue anymore. Can any one help me complete this?
Thanks in advance!