The constant of interpolation (Q1884539)

A sequence \(X=\{z_j\}_{j=1}^\infty\) in \({\mathbb C}_+\) is said to be an interpolating sequence if the interpolation problem \(f(z_j)=w_j\), \(j\in{\mathbb N}\) has a solution \(f\in H^\infty\) for each bounded sequence \(\{w_j\}_{j=1}^\infty\) in \({\mathbb C}\). The constant of interpolation \(M(Z)\) is defined as the smallest \(C>0\) such that \(\| f\| _\infty\leq C\| w\| _\infty\) for all solutions \(f\) of the above interpolation problem. Upper and lower bounds for \(M(Z)\) have been found by \textit{P. Jones} [Acta Math., 150 (1-2), 137-152 (1983; Zbl 0516.35060)] and V.P. Havin in an appendix written to the book \textit{P. Koosis} [Introduction to \(H_p\) spaces. With two appendices by V. P. Havin. 2nd ed. (English) (Cambridge Tracts in Mathematics. 115. Cambridge: Cambridge University Press). (1998; Zbl 1024.30001)]. The authors found a family of estimates for the constant of interpolation \(M(Z)\leq ec_J(Z,g)\) depending on a free parameter \(g\) which is an analytic function on \({\mathbb C}_+\) such that \(| g| \) has a harmonic majorant and \(g(i)=1\). For a special choice of \(g\) this estimate is shown to coincide with the Jones formula. The ''optimal'' estimate \(M(Z)\leq e\inf_{g}c_J(Z,g)\) is shown to be sharp.