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

Lamia Abbas, André Draux

• This paper is devoted to Markov�Bernstein and Landau�Kolmogorov type inequalities in several variables in L2 norm. The measures which are used, are the Hermite measure and some closely connected measures in such a way that the partial derivatives of the orthogonal polynomials are also orthogonal. These orthogonal polynomials in several variables are built by tensor product of the orthogonal polynomials in one variable. These inequalities are obtained by using a variational method and they involve the square norms of a polynomial p and at most those of the partial derivatives of order 2 of p.

  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.