Komatsu has dealt extensively with Gevrey classes of ultra-differentiable functions on an open domain of \(R^ n\) of type s \((s>1)\) and J. Ramis has studied Gevrey classes of functions of type s \((-\infty <s<\infty)\) in case of one point support [Mem. Amer. Math. Soc. 296 (1984; Zbl 0555.47020)]. For \(s\leq 1\), these functions are analytic and usual techniques of functional analysis are not applicable. Here Gevrey classes of ultra-differentiable functions of type s \((-\infty <s<\infty)\) defined on the unit circle T have been studied and these classes of functions have been characterized by the growth conditions on their Fourier coefficients. Since the space T is compact, the method of functional analysis can be used to define and estimate the Fourier coefficients of dual elements, that is, of ultra-distributions of generalized sense.
Two classes of ultradifferentiable functions of type s have been considered - Gevrey-Roumieu class and Gevrey-Beurling class. One of the main results, Theorem 3.3 characterizes the Gevrey-Roumieu ultra- differentiable functions by means of their Fourier coefficients. Another result (Theorem 4.4) deals with the Fourier coefficients of elements of the dual spaces of these classes, that is, of ultra-distributions. These results have been used by the author in further consideration of the spaces of ultra-distributions elsewhere.