Constructive description of a class of periodic functions on the real line

With a help of some family ${\mathcal H}$ of convex nondecreasing functions on $[0, \infty)$ we define the space $G({\mathcal H})$ of $2 \pi$-periodic infinitely differentiable functions on the real line with given estimates for all derivatives. A description of the space $G({\mathcal H})$ is obtained in terms of the best trigonometric approximations and the rate of decrease of the Fourier coefficients. There are given families ${\mathcal H}$ for which functions from $G({\mathcal H})$ can be extended to analytic functions in the horizontal strip of the complex plane. An internal description of the space of such extensions is obtained. Examples of a family of convex functions ${\mathcal H}$ are given.