De la Vallee Poussin problem in the kernel of the convolution operator on the half-plane

V. V. Napalkov ; K. Zimens

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 19 (2015) no. 2, pp. 283-292

Résumé

We consider the multipoint de la Vallee Poussin (interpolational) problem in the half-plane $D$, $D=\{z \, :\, \mathop{\mathrm{Re}} z\alpha,$ $ \alpha>0\}$. Let $\psi(z)\in H(D)$; $\mu_1$, $\mu_2$, $\ldots \in D$ be the positive zero points of this function and let the boundary of domain $D$ contain their limit. Also, we assume that $\mu_k$ is of $s_k$ multiplicity, $k=1, 2, \dots$. Let us set $M_{\varphi}$ an operator of convolution with the characteristic function $\varphi(z)$. Taking an arbitrary sequence $a_{kj},$ $j=0, 1, \ldots, s_k-1$ we should ask: is there a function $u(z) \in \mathop{\mathrm{Ker}}M_\varphi$ that provides the relation $u^{(j)}(\mu_{k})=a_{kj},$ $j=0, 1,\dots,s_k-1$? We assume the operator characteristic function to be of completely regular growth. The solvability conditions for the multipoint de la Vallée Poussin problem in the half-plain and in the bounded convex domains are obtained.

Détail

Keywords: convolution operator, de la Vallee Poussin problem, the half-plane.
Mots-clés : multiple interpolation