Uniform approximation of Poisson integrals of functions from the class H_omega by de la Vallee Poussin sums

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vall\'{e}e Poussin sums on the sets C^{q}_{\beta}H_\omega of Poisson integrals of functions from the class H_\omega generated by convex upwards moduli of continuity \omega(t) which satisfy the condition \omega(t)/t\to\infty as t\to 0. As an implication, a solution of the Kolmogorov-Nikol'skii problem for de la Vall\'{e}e Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H^\alpha, 0<\alpha <1, is obtaine

On partial derivatives of multivariate Bernstein polynomials

It is shown that Bernstein polynomials for a multivariate function converge to this function along with partial derivatives provided that the latter derivatives exist and are continuous. This result may be useful in some issues of stochastic calculus