Interpolating bases in function spaces \(C_\omega^0(K)\) and \(H_\omega^0(K)\) (Q1293579)

Let \(F\) be a Banach space of functions on the set \(K\), and let \(\{x_i\}_{i=0}^{\infty} \subset K\) be a sequence of distinct points. A base \(\{f_i\}_{i=0}^{\infty}\) in \(F\) is called interpolating at the nodes \(\{x_i\}_{i=0}^{\infty}\), if for any function \(f\in F,\) it follows \(f=\sum_{i=0}^{\infty}c_if_i\), and for any number \(n{\geq}0\) there holds \[ \sum_{i=0}^{n}c_if_i(x_j)=f(x_j)\quad \forall~j=0,1,\dots,n. \] In the present paper the author considers two particular Banach spaces of functions on a set \(K{\subset}R\), denoted by \(C^0_{\omega}(K)\) and respectively \(H^0_{\omega}(K)\) which satisfy some special conditions, the set \(K\) being infinite and bounded in the Euclidean metric (i.e., precompact), next for an arbitrary sequence of distinct points \(\{x_i\}_{i=0}^{\infty} \subset K\), dense in \(K\), the author constructs bases in these Banach spaces interpolating at the nodes \(\{x_i\}_{i=0}^{\infty}\). The condition under which the constructed bases are monotone is also studied.