Is the inverse Fourier transform on $L^2$ computable? That is, \begin{align}F^{-1}: &L^2 \rightarrow L^2\\ & f \mapsto \check{f} \end{align} $(\delta_{L^2}, \delta_{L^2})$ computable? Here, we know that $\check{f}(x) = \hat{f}(-x)$ almost everywhere in $L^2(\mathbb{R})$ where $\hat{f}$ is the Fourier Transform of $f$, that is, $\hat{f}(\xi)= \int f(x)e^{-2\pi\iota\xi x}dx$ . We already know that the Fourier Transform $F: L^2\rightarrow L^2$ given by $f \mapsto \hat{f}$ is $(\delta_{L^2}, \delta_{L^2})$ computable by the paper "Type-2 computability on spaces of Integrable functions" by Daren Kunkle.
I think yes because given $\delta_{L^2}$ name of $f$, we can computably find $\delta_{L^2}$ name of $\hat{f}$. Now, if $\lVert \hat{f}-\sum c_j\chi_{(a_j,b_j)}\rVert < 2^{-n}$ then $\lVert \check{f}-\sum c_j\chi_{(-b_j,-a_j)}\rVert < 2^{-n}$ and hence we can get $\delta_{L^2}$ name of $\check{f}$.