On a class of periodic functions in $\mathbb{R}^n$

By means of some family ${\mathcal H}$ of separately radially convex in $\mathbb{R}^n$ functions we define a space $G({\mathcal H})$ of $2\pi$-periodic in each variable infinitely differentiable in $\mathbb{R}^n$ functions with prescribed estimates on all partial derivatives. We describe the space $G({\mathcal H})$ in terms of the Fourier coefficients. We find conditions on the family ${\mathcal H}$, under which the functions from $G({\mathcal H})$ can be continued to functions holomorphic in a tubular domain in $\mathbb{C}^n$. We obtain an inner description of the space of such continuations. The considered problems are directly related with works by P.L. Ul'yanov in the end of 1980s, in which he succeeded to describe completely the classes of $2\pi$-periodic functions of Gevrey type on the real axis not only by the decay rate of the Fourier coefficients but also in terms of the best trigonometric approximations. The obtained results are new both for the case of many variables and the case of a single variable. In particular, the novelty is owing to imposing condition $i_4$) on the family ${\mathcal H}$.