(untitled) (Journal of Computational Mathematics, 2018)
One-dimensional polynomial interpolation does not guarantee the convergency and the stability during numerical computation. For two (or multi)-dimensional interpolation, difficulties are much more raising. There are many fundamental problems, which are left open. In this paper,we begin with the discussion of reproducing kernel in two variables. With its help we deduce a two-dimensional interpolation formula. According to this formula, the process of interpolation will converge uniformly, whenever the knot system is thickened in finitely. We have also proven that the error function will decrease monotonically in the norm when the number of knot points is increased. In our formula, knot points may be chosen arbitrarily without any request of regularity about their arrangement. We also do not impose any restriction on the number of knot points. For the case of multi-dimensional interpolation, these features may be important and essential.