Conditions for the completeness of the system of polynomials

F. S. Lisin

Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 507-513.

Résumé

We consider the space $A_2(K,\gamma)$ of functions which are analytic in the unit disk $K$ and square-summable in $K$ with respect to plane Lebesgue measure $\sigma$ with weight $\gamma=|D|^2$, $D\in A_2(K, 1)$, $D(z)\ne0$, $z\in K$. We establish the inequality $$ \int_K|Dg|^2u\,d\sigma\leqslant\int_ku\,d\sigma, $$ where $g$ represents the distance from $1/D$ to the closure of the polynomials [in the metric of $A_2(K,\gamma)$] and $u$ is any function which is harmonic and nonnegative in $K$. By means of this inequality we obtain sufficient conditions for the completeness of the system of polynomials in $A_2(K,\gamma)$ in terms of membership of certain functions of $D$ in the class $H_2$ (Hardy-2).