Deep convolutional neural networks and data approximation using the fractional Fourier transform

In the first part of this paper, we define a deep convolutional neural network connected with the fractional Fourier transform (FrFT) using the $\theta$-translation operator, the translation operator associated with the FrFT. Subsequently, we study $\theta$-translation invariance properties of this network. Unlike the classical case, these networks are not translation invariant. \par In the second part, we study data approximation problems using the FrFT. More precisely, given a data set $\fl=\{f_1,\cdots, f_m\}\subset L^2(\R^n)$, we obtain $\Phi=\{\phi_1,\cdots,\phi_\ell\}$ such that $$V_\theta(\Phi)=\argmin\sum_{j=1}^m \|f_j-P_{V}f_j\|^2,$$ where the minimum is taken over all $\theta$-shift invariant spaces generated by at most $\ell$ elements. Moreover, we prove the existence of a space of bandlimited functions in the FrFT domain which is ``closest" to $\fl$ in the above sense.