Multiple interpolation by Blaschke products

I. V. Videnskii

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 195-202

Résumé

Basic result: let $\{z_n\}$ be a sequence of points of the unit disc and $\{k_n\}$ be a sequence of natural numbers, satisfying the conditions: $$ \inf_m\prod^\infty_{n=1,n\ne m}\biggl|\dfrac{z_m-z_n}{1-z_nz_m}\biggr|^{k_n}>\delta>0,\quad \sup_n k_n=N+\infty. $$ Then for any bounded sequence of complex numbers $\omega$, $\omega=\{\omega_n^{(k)}\}^{\infty,k_n-1}_{n=1,k=0}$, there exists a sequence $\Lambda=\{\lambda_n^{(k)}\}^{\infty,k_n-1}_{n=1,k=0}$ such that the function $f=M\|\omega\|_{\infty}B_\Lambda$ interpolates $\omega$: $$ f^{(k)}(z_n)(1-|z_n|^2)^k/K!=\omega_n^{(k)}, $$ where $B_\Lambda$ is the Blaschke product with zeros at the points $\{\lambda_n^{(k)}\}$, $M$ is a constant, $|M|31^N/\delta^N$, $|\lambda_n^{(k)}-z_n|/|1-\overline{\lambda}_n^{(k)}z_n|\delta/31^N$. If $N=1$ this theorem is proved by Earl (RZhMat, 1972, IB163). The idea of the proof, as in Earl, is that if the zeros $\{\lambda_n^{(k)}\}$ run through neighborhoods of the points $z_n$, then the Blaschke products with these zeros interpolate sequences $\omega$, filling some neighborhood of zero in the space $l^\infty$. The theorem formulated is used to get interpolation theorems in classes narrower than $H^\infty$.