Interpolation by derivatives in \(H^{\infty}\)
One of the main problems studied in the paper concerns the simultaneous interpolation by the derivatives \(f^{(n)}\) and \(f^{(m)}\), of a bounded, analytic function \(f\) on the unit disk. Given a sequence \(C=(c_k)\) of points of the disk, the authors introduce a weighted \(\ell^\infty\) class of complex sequences \[ \Lambda_k(C):=\{(\lambda_j)_{j=1}^\infty: \quad \sup_j | \lambda_j| (1-| c_j| ^2)^k<\infty \}. \] Given two disjoint sequences \(A=(a_j)\), \(B=(b_j)\) in the unit disk, their union \(Z=A\cup B\) is said to be \((n,m)\)-interpolating with respect to \((A,B)\) if for each \((\alpha_j)\in\Lambda_n(A)\) and \((\beta_j)\in\Lambda_m(B)\) there is a function \(f\in H^\infty\) so that \[ f^{(n)}(a_j)=\alpha_j, \qquad f^{(m)}(b_j)=\beta_j, \qquad j=1,2,\ldots. \] The main result of the paper claims that \(Z=A\cup B\) is \((n,m)\)-interpolating with respect to \((A,B)\) if and only if both \(A\) and \(B\) are uniformly separated.
scientific article; zbMATH DE number 6837282
scientific article