A MULTIVARIATE FORM OF HARDYS INEQUALITY AND L(P)-ERROR BOUNDS FOR MULTIVARIATE LAGRANGE INTERPOLATION SCHEMES

The multivariate generalization of Hardy's inequality-that for m - n/p > 0, [graphics] valid for f is an element of L(p)(IR(n)) and Theta an arbitrary finite sequence of points in IR(n)-is discussed. The linear functional f --> integral(Theta)f was introduced by Micchelli in conn ection with Kergin interpolation. This functional also naturally occur s in other multivariate generalizations of Lagrange interpolation, inc luding Hakopian interpolation and the Lagrange maps of section 5. For each of these schemes, ()implies L(p)-error bounds. We discuss why (* )plays a crucial role in obtaining L(p)-bounds from pointwise integral error formulas for multivariate generalizations of Lagrange interpola tion.