Asymptotic behavior for the discrete in time heat equation

In this paper, we investigate the asymptotic behavior and decay of the solution of the discrete in time N-dimensional heat equation. We give a convergence rate with which the solution tends to the discrete fundamental solution, and the asymptotic decay, both in Lp(RN). Furthermore, we prove optimal L2-decay of solutions. Since the technique of energy methods is not applicable, we follow the approach of estimates based on the discrete fundamental solution which is given by an original integral representation and also by MacDonald’s special functions. As a consequence, the analysis is different to the continuous in time heat equation and the calculations are rather involved

Spectral sets of generalized Hausdorff matrices on spaces of holomorphic functions on D

Here, we study a family of bounded operators H, acting on Banach spaces of holomorphic functions X→O(D), which are subordinated in terms of a C0--semigroup of weighted composition operators (vtCϕt ), i.e.,H= ∞ 0 vtCϕt dν(t) in the strong sense for some Borel measure ν. This family of operators extends the so-called generalized Hausdorff operators. Here, we obtain the spectrum, point spectrum and essential spectrum of H under mild assumptions on (vtCϕt ),ν and X. In particular, we obtain these spectral sets for a wide family of generalized Hausdorff operators acting on Hardy spaces, weighted Bergman spaces, weighted Dirichlet spaces and little Korenblum classes. The description for the spectra of the infinitesimal generator of (vtCϕt) is the key for our findings

C0-semigroups and resolvent operators approximated by Laguerre expansions

In this paper we introduce Laguerre expansions to approximate vector-valued functions. We apply this result to approximate C0- semigroups and resolvent operators in abstract Banach spaces. We study certain Laguerre functions in order to estimate the rate of convergence of these expansions. Finally, we illustrate the main results of this paper with some examples: shift, convolution and holomorphic semigroups, where the rate of convergence is improved