  In this short note we have proved an enhanced version of a theorem of Lorentz
[1] and its generalization to the multivariate case which gives a non- uniform
estimate of degree of approximation by a polynomial with positive coefficients.
The performance of the approximation at the vertices of [0; 1]d is more
precisely characterized by the improved result and its multivariate
generalization. The latter provides mathematical foundation on which
multivariate density approximation by a polynomial with positive coefficients
can be established.
