On optimal recovery of a holomorphic function in the unit ball of \(\mathbb{C}^ n\) (Q1185193)

Let \(\Omega\subseteq\mathbb{C}^ n\), let \(\mu\) be a positive measure on \(\Omega\) and \(1\leq p\leq\infty\). Let \(X_ p\) be a linear subspace of \(L_ p(\Omega,\mu)\) with the induced norm and \(BX_ p\) be the unit ball of \(X_ p\). An information operator \(I\) is a mapping \(I: BX_ p\to Y\), \(Y\) being an arbitrary set. Let \(S: Y\to C\) be a function on \(Y\) and \(L\) be a functional on \(X_ p\). The authors consider the problem of optimal recovery, i.e. to determine: \[ E(L,I,X_ p)=\inf_ S \sup_{f\in BX_ p}| Lf-SIf|. \] If \(A\subseteq\Omega\), \(a\in\Omega\setminus A\), \(L(f)=f(a)\), \(I(f)=f\mid_ A\) and \(X_ p\) is the space of holomorphic functions in \(\Omega\) which belong to \(L_ p(\overline{\Omega},\mu)\) then under some conditions the authors obtain the exact form of \(E(L,I,X_ p)\), which is expressed using the reproducing kernels in some weighted spaces. This general result is applied to the recovery problem in the Hardy and Bergman spaces. In particular one obtains generalizations of the Schwartz lemma for functions from Hardy and Bergman spaces.