Let $$\Psi_{J,K,\theta}(x,y)=\frac{1}{\sqrt{J}}\Psi\left(\frac{\cos(\theta)x+\sin(\theta)y-K}{J}\right)$$ Let $$\mathcal{T}[f](J,K,\theta)=\int_{\mathbb{R}^2}\overline{\Psi_{J,K,\theta}(x,y)}f(x,y)\:dxdy$$
(First claim that i dont understand) Then by Parseval equality we get $$=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\overline{\hat{\Psi}_{J,K,\theta}(\xi)}\hat{f}(\xi)\:d\xi$$ Isn't the Paserval equality something related to the square modulos of a function and its corresponding fourier transform?
(Here i believe that are some erros but im sure about it) By writing $\xi=\xi(\theta,R)=(R\cos(\theta),R\sin(\theta))$ we get $$=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\sqrt{J}\:\overline{\hat{\Psi}(JR)}e^{-iRK}\hat{f}(\xi(\theta,R))\:dR$$ Here my doubts are related to the expoent of the exponential and with the differential. Shouldn't we have $e^{-iR\frac{K}{J}}$ and shouldn't we consider a diferential for $\theta$ too?
Thank you for your attention. Best Regards.