Interpolation of functions of the Gevrey class in the closed disk and ball (Q1174008)

Let \(\Omega_ r=\{z\in\mathbb{C};\;| z|<r\}\). The function \(f\) is contained in the Gevrey class \(C^ s(\Omega_ 1)\), \(s\geq 0\), if \(f\) is holomorphic in \(\Omega_ 1\), infinitely differentiable in \(\overline\Omega_ 1\) and \(\max_{z\in\overline\Omega_ 1}\left|{\partial^ k \over \partial z^ k}f(z) \right|\leq M_ f C^ k_ f(k!)^{1+s}\), for \(k=0,1,2,\dots,\) where constants \(M_ f\) and \(C_ f\) depend on \(f\). Let \(\{a_ j\}\subset\overline\Omega_ 1\), \(b_ 1,\dots,b_ s\) denote all distinct representatives from \(a_ 1,\dots,a_ m\), and \(\alpha_ j\) denote the multiplicity of \(b_ j\) in \(a_ 1,\dots,a_ m\), where \(\sum_{j=1}^ s \alpha_ j=m\). The interpolation functions \(f\in C^ s(\overline\Omega_ 1)\) with knots \(a_ 1,\dots,a_ m\) and rational functions with poles in \(1/t_ m \bar a_ 1,\dots,1/t_ m \bar a_ m\), \(0<t_ m<1\), is considered. A linear interpolation operator is defined as follows: \[ L_ m(f(z))=f_ m(z)=P_ m(z)/\tilde A(z), \] where \(P_ m(z)\) is a Hermitian polynomial, and \(\tilde A(z)=\prod_{j=1}^ s (t_ m b_ j-1)^{\alpha_ j}\). In this paper, in particular, the following theorem is proved. Theorem 1. If \(f\in C^ \infty(\overline\Omega_ 1)\), \(s<1\), and \(t_ m=1-2/\kappa_ m\), where \(\kappa_ m=o(m)\) and \(m^ s=o(\kappa_ m)\) as \(m\to\infty\) then \(f_ m\) is uniformly convergent to \(f\) in \(\overline\Omega_ 1\).