Lidstone approximation on the triangle

In this paper we give a contribution to Lidstone approximation [Proc. Edinburgh Math. Soc. 2 (1929) 16] with a new approximation formula on the simplex. Interpolation conditions satisfied by the proposed formula are studied and sufficient conditions for the uniform convergence are given. In particular, a class of functions for which the sequence of approximating polynomials uniformly converges to function is determined, by extending to the class of two variate ridge functions [Cheney and Light, A Course in Approximation Theory, Brooks/Cole Publishing Company, 1999] the classic notion of completely convex function [Bull. Amer. Math. Soc. 47 (1941) 750; Trans. Amer. Math. Soc. 51 (1942) 387]. Finally, it is shown that the calculation of the approximating polynomial can be organized in a stable algorithm and some numerical test are performed.