For n 1, let fxjngnj=1 be n distinct points in a compact set K R and let Ln[ ] denote the corresponding Lagrange Interpolation operator. Let v be a suitably restricted function on K. What conditions on the array fxjng1 j n; n 1 ensure the existence of p > 0 such that lim n!1 k (f Ln[f ]) v kLp(K)= 0 for every continuous f :: K ! R ? We show that it is necessary and su cient that there exists r ...

In this paper, we derive an explicit formula for the bivariate Lagrange basis polynomials of a general set checkerboard nodes. This generalizes existing results at Padua nodes, Chebyshev Morrow-Patterson and Geronimus We also construct subspace spanned by linearly independent vanishing that vanish nodes prove uniqueness in quotient space defined as with certain degree over polynomials.