We consider a function $f$ belonging to $\mathcal{B}_1$ and we recall that $$\hat{f}(q,\theta) = \int_{-\infty}^{+\infty} f(q\cos\theta - t\sin\theta,\, q\sin\theta + t\cos\theta)\,\mathrm{d}t$$ Verify that $\hat{f}$ is defined on $\mathbb{R}^2$.
We consider a function $f$ belonging to $\mathcal{B}_1$ and we recall that
$$\hat{f}(q,\theta) = \int_{-\infty}^{+\infty} f(q\cos\theta - t\sin\theta,\, q\sin\theta + t\cos\theta)\,\mathrm{d}t$$
Verify that $\hat{f}$ is defined on $\mathbb{R}^2$.