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4 papers
Short communications
Multiple interpolation by Blaschke products
I. V. Videnskii
Abstract:
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
$H^\infty$.
UDC:
517.948:513.8, 519.4