A Remainder Formula for Hermite Multivariate Interpolation
by
Dana Simian
"Lucian Blaga" University of Sibiu, Romania

The aim of this paper is to present and prove a remainder formula in two cases of Hermite interpolation in two variables and to apply this formula to certain particular Hermite interpolation spaces.
The two Hermite interpolation cases we study in this article are minimal interpolation spaces with respect to a set of functionals, \Lambda, which more important property is: ker(\Lambda) is a polynomial ideal.
In the first Hermite interpolation case considerated, the conditions \Lambda satisfy the equality:

span(\Lambda) = span{ \deltaxj; xj in R2; j=1, ... , m; xi =/= xj  for all i =/= j}

We denoted by \delta_xj the evaluation functional on the point x_j.
Hence

\lambda(f)= N å j=1  cj\lambda ·\deltaxj(f),

the real coefficients c_j^\lambda are \lambda -dependents for each \lambda in \Lambda.
The main results we obtain for this choise of functionals are given by the expressions of (f-W_n(f))(x) and (g(D)(W_n(f)))(x), withW_n(f) the Hermite interpolation operator, and g(D) the differential operator with constant coefficients associated to the bivariate polynomial g.
Another choice of the set of functionals is:
\Lambda = {\lambda_{j, k}:j=1, ... , m; k=0, ... , l_j-1}, with

\lambdaj, k=\deltaxj o qj, k(D) ,  qj, k = x\alphaj, k,  \alphaj, k in N2,  x in R2, Qj =span { qj, k : k=0, ... , lj-1} is D- invariant,   m å j=1  lj=N

We puted the set \Lambda into blocks and derived the Newton basis in a way which is able to characterize the special structure of this functionals set. In this way, we obtained the formula of the remainder in this interpolation case.
Another result consists in finding superior bounds for the remainder in the Hermite interpolation cases we studied.
In the end, we applied these formulas to derive the interpolation remainder for the interpolation with weights and least interpolation.

Date received: July 11, 2003