Let $f\in L^1(\Bbb{R})$ and define by $\hat f(\xi)=\int_{\Bbb{R}}f(x)e^{-i\xi x}d\mu(x)$ the Fourier transform of $f$. Prove: $\hat f(\xi)=\frac{1}{2}(\widehat {f-f_{\frac{\pi}{\xi}}})(\xi)$, where $f_{\frac{\pi}{\xi}}=f(x-\frac{\pi}{\xi})$.
My attempt: Let $f\in L^1(\Bbb{R})$. Then:
$$(\widehat {f-f_{\frac{\pi}{\xi}}})(\xi)=\int_{\Bbb{R}}(f(x)-f_{\frac{\pi}{\xi}}(x))e^{-i\xi x}d\mu(x)=\int_{\Bbb{R}}f(x)e^{-i\xi x}d\mu(x)-\int_{\Bbb{R}}f(x-\frac{\pi}{\xi})e^{-i\xi x}d\mu(x)$$
Now if I could use the change of variables $x-\frac{\pi}{\xi}\to y$, I would get the equality I want. But I don't know if I can use change of variables in the case of Lebesgue integrals. Any help would be appreciated.