Basicity of some biorthogonal systems and the solution of a~multiple interpolation problbm in the $H^p$ classes in the half-plane

M. M. Dzhrbashyan

Izvestiya. Mathematics , Tome 13 (1979) no. 3, pp. 589-646

Résumé

Let $\{\lambda_k\}_1^\infty$ be a sequence in $G^{(+)}=\{z:\operatorname{Im}z>0\}$, and $s_k$ the multiplicity of the occurrences of $\lambda_k$ in the segment $\{\lambda_1,\dots,\lambda_k\}$. Also let $H_+^p$ $(1$ be the space of functions $f(z)$ holomorphic in $G^{(+)}$ that obey $$ \|f\|_p=\sup_{0+\infty}\biggl\{\int^{+\infty}_{-\infty}|f(x+iy)|^p\,dx\biggr\}^{1/p}\infty. $$ The article gives a completely internal characterization of systems of the form $\bigl\{r_k(z)=\frac{(s_k-1)!}{(z-\overline\lambda_k)^{s_k})}\bigr\}^\infty_{k+1}$ that are not total in $H^p_+$ and of the biorthogonal systems $\{\Omega_k(z)\}_1^\infty$ constructed for such nontotal systems. The closed linear hulls of the systems $\{r_k(z)\}_1^\infty$ and $\{\Omega_k(z)\}_1^\infty$ are also characterized. Criteria for these systems to be bases in their closed linear hulls in the metric of $H^p_+$ are obtained. A complete and effective solution of the multiple interpolation problem in the classes $H_+^p$ is given. In addition it is proved that functions with given interpolation data can be represented both as an interpolation series in the system $\{\Omega_k(z)\}_1^\infty$ and as a series in the system $\{r_k(z)\}_1^\infty$. Bibliography: 20 titles.