A survey of the theory of operator polynomial interpolation - Makarov, V.; Kashpur, O. (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 517.988 The present survey is devoted to the 85th birthday of Volodymyr Volodymyrovych Khlobystov, prominent Ukrainian mathematician, renowned for his pioneering contributions to computational mathematics. He was a unique scholar who, despite severe visual impairments (almost total blindness) managed to get leading results in the theory of operator polynomial interpolation. Together with his colleagues and numerous students, he laid the foundations of the contemporary theory of operator polynomial interpolation. The present survey, as compared with the most well-known works of other researchers, highlights a much greater depth and power of Khlobystov’s results. As the most important of his results, we can mention the creation of a new direction in interpolation theory based on continuous nodes. For the first time, this approach balanced the continuous amount of information about the interpolated object with the continuous set of interpolation nodes in the proposed interpolants. The operator interpolation problems of Lagrange, Hermite, and Hermite-Birkhoff types were solved, the conditions for the existence and uniqueness of solutions were established, the constructive description of the entire set of corresponding interpolants was provided, a subset of interpolation polynomials preserving the polynomials of a given degree was selected, the accuracy of the interpolation formulas was analyzed, and the convergence of the interpolation processes was investigated.