Consider, for a fixed $n \geq 0$, the "Hankel-type" transform $$\hat{f}(k)=(2\pi)^{-(n+2)/2}\int_0^\infty dx\,x^{n+1}\left(\int_0^\pi d \theta\,\sin^n(\theta) e^{ikx\cos(\theta)}\right)f(x)$$ a priori defined on the weighted space $L^1(\mathbb{R}^+,x^{n+1})$. Can one pull off a straightforward ad hoc proof that for any $f \in L^1(\mathbb{R}^+,x^{n+1}) \cap L^2(\mathbb{R}^+,x^{n+1})$ we have $$\int_0^\infty dk \,k^{n+1}|\hat{f}(k)|^2 = \int_0^\infty dx \,x^{n+1}|f(x)|^2$$ ?
(For integer $n$, the result follows from Parseval's theorem for the Fourier transform in $\mathbb{R}^{n+2}$ applied to radially symmetric functions)