On a Bivariate Interpolation Problem Original Research Article

In this paper we solve the poisedness problem for a bivariate interpolation introduced by B. Bojanov and Y. Xu. The authors were informed the problem was solved earlier, by a different method, also by B. Bojanov and Y. Xu (to appear, On a Hermite interpolation by polynomials of two variables, SIAM J. Numer. Anal.). Parameters of the original interpolation from Π2k are the values of a function and its radial derivatives up to some order k at 2k+1 equidistant nodes of the unit circumference. We also consider other closely related polynomial interpolations and prove their poisedness. Meanwhile the poisedness of several general univariate Birkhoff interpolation problems is proved to which the above problem is reduced. At the end we consider the corresponding useful cubature formulas.