Regularity of tensor product approximations to square integrable functions (Q662815)

Given a function \(f\in L^2({\mathbb R}^{d_1+\dots +d_N})\) it may be impossible to solve the problem \[ \left \|f - \sum_{k=1}^r u_k^1\otimes\dots\otimes u_k^N\right \|_{L^2} = \min, \eqno (1) \] for \(u_k^n\in L^2({\mathbb R}^{d_n})\). The author focuses on local minima of (1) and analyzes the corresponding regularized minimization problem. He also investigates the approximation of \(f\) by a tensor of the so-called Tucker format or rank-\((r_1,\dots,r_N)\) approximation: \[ \left \|f - \sum_{{k_1}=1}^{r_1}\dots\sum_{{k_N}=1}^{r_N} \alpha_{k_1,\dots,k_N} u^1_{k_1}\otimes\dots\otimes u^N_{k_N}\right \|_{L^2} = \min. \eqno (2) \] The integers \(r_1,\dots, r_N>0\) are given, the functions \(u_k^n\in L^2({\mathbb R}^{d_n})\) and the coefficients \(\alpha_{k_1,\dots,k_N}\) are to be determined. For the spaces \(H^s\), \(C_0^m\), and \({\mathcal S}\) it is shown that the best approximation and all of its factors in (1) and (2) have the same smoothness as the approximated function itself.