Envelopes, widths, and Landau problems for analytic functions (Q1119850)

Denote by \(A_ r\) the set of all analytic functions on the open unit disk \(D=\{z:\) \(| z| <1\}\) which are real-valued on the interval (-1,1) and satisfy the condition \(| f^{(r)}z| \leq 1\) for \(z\in D\). Let \(x_ 0,x_ 1,...,x_{n+r}\) be points from (-1,1). Put \(M=\{f(x_ 0),...,f(x_{n+r}):\) \(f\in A_ r\}\). Using the Blaschke product \(B(z)=\lambda \prod^{m}_{j=1}(z-a_ j)/(1-\bar a_ jz)\) \((| \lambda | =1\); \(a_ 1,...,a_ m\in D)\) the author gives a characterization of boundary points of M. It can be deduced from this that a single function provides an ``envelope'' for the graph of f(x), - 1\(\leq x\leq 1\), \(f\in A_ r\) and \(f(x_ j)=0\) \((j=1,2,...,n+r)\). If E is a subset of a Banach space X then the Kolmogorov n-width \(d_ n(E;X)\) of E is defined as \(d_ n(E;X)=\inf_{X_ n}\sup_{f E}\inf_{g X_ n}\| f-g\|,\) where \(\| \|\) is the norm of X and \(X_ n\) runs over all n-dimensional subspaces of X. In the paper the Kolmogorov widths of \(A_ r\) in the space of continuous complex-valued functions on a compact set \(F\subset (-1,1)\) are determined.