On the space of periodic ultradifferentiable functions of Roumieu type and its dual

With a help of a family ${\mathcal H}$ of convex nondecreasing functions on $[0, \infty)$ we define the space $J({\mathcal H})$ of $2 \pi$-periodic infinitely differentiable functions on the real line with given estimates for all derivatives. It belongs to the class of spaces of ultradifferentiable functions of Roumieu type. 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. A general form of linear continuous functionals on $J({\mathcal H})$ is found. It is shown that some well-known classes of $2 \pi$-periodic functions of Gevrey type are special cases of the spaces $J({\mathcal H})$.