On the Space of Periodic Ultradifferentiable Roumieu Functions and Its Dual Space

Using family ℋ of convex functions nondecreasing on [0,∞ ) , the space J(ℋ) of 2π -periodic functions infinitely differentiable on a straight line with given estimates on derivatives is defined. Its description is obtained in terms of the best trigonometric approximations of functions and the rate of decay of their Fourier coefficients. The general form of linear continuous functionals on J(ℋ) is found. It is shown that some known classes of 2π -periodic Gevrey functions are special cases of J(ℋ) spaces.