The completeness of systems of functions of the Mittag--Leffler type for weighted uniform approximation in a~complex

I. O. Khachatryan

Matematičeskie zametki, Tome 18 (1975) no. 5, pp. 675-685.

Résumé

For a given $\rho$ ($1/2\rho+\infty$) let us set $L_\rho=\{z:|\arg z|=\pi/(2\rho)\}$ and assume that a real valued measurable function $\varphi(t)$ such that $\varphi(t)\ge1$ ($t\in L_\rho$) and $\lim\limits_{|t|\to+\infty}\varphi(t)=+\infty$ $(t\in L_\rho)$ is defined on $L_\rho$. Let $C_\varphi(L_\rho)$ denote the space of continuous functions $f(t)$ on $L_\rho$ such that $\lim\frac{f(t)}{\varphi(t)}=0$, where the norm of an elementf is defined as: $\|f\|=\sup\limits_{t\in L_\rho}\frac{|f(t)|}{\varphi(t)}$. In this note we pose the question about the completeness of the system of functions of the Mittag-Leffler type $\{E_\rho(ut;\mu)\}$ ($\mu\ge1$, $0\le u\le a$) or, what is the same thing, of the system of functions $p(t)=\int_0^aE_\rho(ut;\mu)\,d\sigma(u)$ in $C_\varphi(L_\rho)$. The following theorem is proved: The system of functions of the Mittag-Leffler type is complete in $C_\varphi(L_\rho)$ if and only if $\sup|p(z)|\equiv+\infty$, $z\in L_\rho$, where the supremum is taken over the set of functions $p(t)$ such that $\|p(t)(t+1)^{-1}\|\le1$.