ON 2-DIMENSIONAL POLYNOMIAL INTERPOLATION

A tuple n = {n1 , n(k); n} of positive integers with SIGMA(nu=1)k n(nu )(n(nu) + 1) = (n + 1)(n + 2) is said to be regular if there exists a set U = {u1 , . . . , u(k)} subset-of R2 such that the Hermite interpo lation problem (n, U) is regular, i.e., for arbitrary numbers lambda(i ,j), nu), i + j < n(nu) , nu = 1, . . . k, there exists a unique polyn omial P(x, y) is-an-element-of pi(n)(R2) such that partial derivative( i+j)/partial derivative x(i) partial derivative y(j)P(x,y)\u(nu) = lam bda(i,j),nu, 1 + j < n(nu), nu = 1 , . . . , k. In this paper an algor ithm is obtained that completely describes the regular and singular tu ples n under the condition that n10 = 1 . In the case when only the de rivatives of order n(nu) are interpolated, necessary and sufficient co nditions are obtained for an arbitrary tuple n to be regular. Bibliogr aphy: 9 titles.