Développement en série de fonctions holomorphes des fonctions d'une classe de Gevrey sur l'intervalle [-1;1]

Abstract

We characterize Gevrey functions on the unit interval [−1, 1] as sums of holomorphic functions in specific neighborhoods of [−1, 1]. As an application of our main theorem, we perform a simple proof for Dyn'kin's theorem of pseudoanalytic extension for Gevrey classes on [−1, 1].

Solvability in Gevrey Classes of Some Nonlinear Fractional Functional Differential Equations

“…Our purpose in this paper is to prove, under some regularity conditions on the data, the solvability in a Gevrey class of the form G k,− 1 ([− 1, 1]) of a class of nonlinear FFDE. Our approach is mainly based on a theorem that we have proved in [41]. e notion of fractional calculus we are interested in is the Caputo fractional calculus.…”

Solvability in Gevrey classes of some nonlinear fractional functional differential equations

“…Our purpose in this paper is to prove, under some regularity conditions on the datas, the solvability in a Gevrey class of the form G k,−1 ([−1, 1]) of a class of nonlinear FFDE. Our approach is mainly based on a theorem that we have proved in ( [10]). The notion of fractional calculus we are interested in is the Caputo fractional calculus.…”

“…[10]), that u |[d,1] belongs to the Gevrey class G k ([d, 1]). But since d is an arbitrary element of ]−1, 1[ and u is of class C 1 on [−1, 1] , it follows that u belongs to the Gevrey class G k,−1 ([−1, 1]).…”

Holomorphic series expansion of functions of Carleman type on the interval [−1, 1]

“…Improving the methods of Ecalle and Belghiti, we obtained in [5] a characterization of the functions of a Gevrey class on [−1, 1] as sums of series of holomorphic functions in suitable neighborhoods of [−1, 1], and here we generalize this method to some Carleman classes on [−1, 1]. As an application of our main theorem, we derive an alternative construction of Dyn'kin's pseudoanalytic extension for these Carleman classes.…”