A note on the optimal recovery of functions in \(H^{\infty}\)
Let \(H^{\infty}\) be the Banach space of bounded analytic functions in the disk \(D=\{z:\) \(| z| <1\}\) with norm \(\| f\|_{\infty}=\sup \{| f(z)|\); \(z\in D\}\). Let \([a,b]\) be a given subinterval of \(]-1,1[\). By \(\| f\|\) is denoted the uniform norm of f in [a,b]. This paper is concerned with the problem of best approximations \(f_ N\) from \(H^{\infty}\) on the basis of the data \((\ell_ 1(f),...,\ell_ N(f))\), where \((\ell_ k)_ 1\) are fixed continuous linear functionals. In the case of Hermitian data \(\{f^{(\lambda)}(x_ k)\), \(k=1,...,n\), \(\lambda =0,...,\nu_ k-1\}\) the error of the best approximation of function \(f\in H^{\infty}\) is expressed by the Blaschke product \(B(\bar x,t)\) with zeros \(\bar x=(x_ 1,...,x_ n)\) of multiplicities \(\nu_ 1,...,\nu_ n\) respectively. For given \((\nu_ k)^ n_ 1\) the uniqueness of the nodes \(\bar x^*\) which are optimal of type \((\nu_ 1,...,\nu_ n)\), i.e. which minimize the uniform norm of B(\=x,t) in [a,b] over \(a\leq x_ 1\leq...\leq x_ n\leq b\), is proved.
scientific article; zbMATH DE number 4065593
scientific article