Convergent Lagrange interpolation polynomials for the disk algebra (Q5943696)

Let \({\mathcal A}(D)\) denote the complex disc algebra consisting of the complex-valued functions \(f\) holomorphic on the open unit disc \(D \hookrightarrow {\mathbb C}\) and admitting a continuous extension to the closure \(\overline{D} = D \cup \partial D \in {\mathbb C}.\) For an infinite triangular interpolation matrix \({\mathcal K}\) of knots located on the boundary of the unit circle \(\partial D = {\mathbf S}_1\) and admitting of the sequence of knot polynomials \((\omega_n)_{n \geq 0}\), the author establishes a convergence theorem for the Lagrange interpolation polynomials \(L_{\mathcal K}(f)\) of \(f \in {\mathcal A}(D)\), uniformly valid on the compact subsets of \(D\). The convergence theorem is based on uniform boundedness conditions on the compact subdiscs of \(D\), in terms of the knot polynomials \((\omega_n)_{n \geq 0}\) associated with \({\mathcal K}\) and the reciprocals of the derivatives \((\omega_n')_{n \geq 0}\) evaluated at the knots of \({\mathcal K}\).