Polynomial approximation in local Dirichlet spaces

Abstract: The partial Taylor sums $S_n$, $n \geq 0$, are finite rank operators on any Banach space of analytic functions on the open unit disc. In the classical setting of disc algebra, the precise value of the norm of $S_n$ is not known and thus in the literature they are referred as the Lebesgue constants. In this setting, we just know that they grow like $\log n$, modulo a multiplicative constant, as $n$ tends to infinity. However, on the weighted Dirichlet spaces $\D_w$, we precisely evaluate the norm of $S_n$. As a matter of fact, there are different ways to put a norm on $\D_w$. Even though these norms are equivalent, they lead to different values for the norm of $S_n$, as an operator on $\D_w$. We present three different norms on $\D_w$, and in each case we try to obtain the precise value of the operator norm of $S_n$. These results are in sharp contrast to the classical setting of the disc algebra. We also consider the problem for the cesaro means $\sigma_n$ on local Dirichlet spaces and try to find the norm of $\sigma_n$ precisely for the three different norms that we introduced.

analysis of PDEsclassical analysis and ODEsfunctional analysisprobabilityspectral theory

Audience: researchers in the discipline

Quebec Analysis and Related Fields Graduate Seminar