Moment problems for bounded functions
Let \((X,\mu)\) be a measure space and \(\{ \varphi_j\}\) be a sequence of functions in \(L^1 (X, \mu)\) normalized by \[ |\varphi_j |_j= \int|\varphi_j |d\mu=1. \] The main result of the paper describes a sufficient condition which guarantees that for any sequence \(\{a_j \}\in l^\infty\) there exists \(f\in L^\infty (X, \mu)\) for which \[ \int f\varphi_j d\mu= a_j, \qquad j=1, 2,\dots\;.\tag{1} \] The condition says: there is a collection \(\{E_j \}^\infty_{j =1}\) of pairwise disjoint measurable sets \(E_j \subset X\) such that \[ \sigma= \inf_j \int_{E_j} |\varphi_j |d\mu> 1/2. \] Moreover the solution \(f\) of (1) satisfies estimation \[ |f|_\infty\leq (2\sigma- 1)\sup_j |a_j|. \] The paper also contains several interesting applications of this abstract theorem to interpolation problems for bounded harmonic functions in \(R_+^{n+1}\), to estimates of the growth of entire functions represented by lacunary series and to interpolation problems for BMOA class in the unit disc. These applications and other ones presented in the work illustrate the power of general interpolation theorem.
scientific article; zbMATH DE number 827485