We present the results of the numerical simulation of the first stages of the melting from a side of a gallium slab by adding to the heat transfer and to the melt flow the description of the effects of the deformations of the solid phase. The experiment by Gau and Viskanta in [4] has been considered.
Starting from a natural generalization of the trigonometric case, we construct a de la Vall\'ee Poussin approximation process in the uniform and L1 norms. With respect to the classical approach we obtain the convergence for a wider class of Jacobi weights. Even if we only consider the Jacobi case, our construction is very general and can be extended to other classes of weights.
