On the norm of the Laplace operator on spaces of polynomials
In joint work with Peter Dörfler, we derived Markov-type inequalities for certain partial differential operators on multivariate polynomials with Hilbert space norms. Unfortunately, these results are not applicable to the most important partial differential operator, the Laplace operator. The talk is devoted to first insights gained for just this operator.
We consider the Laplace operator on the finite-dimensional linear space of algebraic polynomials in k variables such that each variable occurs at most with the power n. The space of the polynomials is equipped with the Laguerre norm. We establish safe lower and upper bounds for the norm of the Laplace operator on this space, and we derive asymptotic lower and
upper bounds for this norm as n goes to infinity. The asymptotic bounds are better than the safe bounds.
The talk is based on joint work with Christian Rebs.