Markov-Bernstein and Landau-Kolmogorov type inequalities in several variables for the Hermite and closely connected measures

It is also proved that the sequences of polynomials in several variables, orthogonal with respect to the Hermite and Laguerre-Sonin measures, are closed in a certain Sobolev space regarding its natural inner product and the inner product linked to the bilinear functional which is used in the variational method. This result shows us that the given Landau-Kolmogorov type inequalities are satisfied not only for polynomials, but also for functions belonging to this Sobolev space.