This question deals with Fourier inversion in $\mathbb{R}^n$. Assume $f$ is a function supported on a ball of radius $2R$. Define $\beta_{R}$ to be the normalized average of $f$ over a a ball of radius $R$. Assume $f$ is nice (smooth, Schwartz?). By means of Fourier inversion we have $$f(x) = \int_{z\in\mathbb{R}^{n}}\hat{f}(z)e(<x,z>)dz$$ where the Fourier transform $\hat{f}(z)$ is defined by $$\hat{f}(z)=\int_{x\in\mathbb{R}^n}f(x)e(-<x,z>)dx.$$
Consider now a function $f$ which is positive, and satisfy $\beta_{R}f(x) \ll_{f} R^{-\alpha}$ for some $\alpha>0$. In particular it implies that $\lvert \hat{f}(z)\rvert \ll_{f} R^{n-\alpha}$ for all $z$.
I would like to go the other way, starting from the Fourier transform data and recover the physical data. Which mean to apply $\beta_{R}$ in the Fourier inversion formula and recover the formula $$\beta_{R}f(x) = \int_{z\in\mathbb{R}^{n}}\hat{f}(z)\left(J_{n/2}(R \cdot z)/\lVert R\cdot z\rVert^{n/2}\right)dz$$ where $J_{n/2}$ is the appropriate Bessel function.
My question is - why this integral even converges? not to mention to get an estimate ''related'' to $R^{-\alpha}$? The normalization of the kernel is $L^{2}$ hence I am ''missing'' half of the dimensions required for the normalization to hold.