TT ranks of approximate tensorizations of some smooth functions

Tensorizations of functions are studied, that is, tensors with elements $A(i_1,\dots,i_d)=f(x(i_1,\dots,i_d))$, where $f(x)$ is some function defined on an interval and $\{x(i_1,\dots,i_d)\}$ is a grid on this interval. For tensors of this type, the problem of approximation by tensors admitting a tensor train (ТТ) decomposition with low ТТ ranks is posed. For the class of functions that are traces of analytic functions of a complex variable in some ellipses on the complex plane, upper and lower bounds for ТТ ranks of optimal approximations are deduced. These estimates are applied to tensorizations of polynomial functions. In particular, the well-known upper bound for ТТ ranks of approximations of such functions is improved to $O(\log n)$, where $n$ is the degree of the polynomial.