Anisotropic Franklin bases on polygonal domains

Abstract

Franklin systems induced by Courant elements over multilevel nested triangulations of polygonal domains in ℝ2 are explored. Mild conditions are imposed on the triangulations which prevent them from deterioration and at the same time allow for a lot of flexibility and, in particular, arbitrarily sharp angles. It is shown that such anisotropic Franklin systems are Schauder bases for C and L1, and unconditional bases for Lp (1 < p < ∝ ) and the corresponding Hardy spaces H 1. It is also proved that the anisotropic H1 is exactly the space of all functions in L1 for which the corresponding Franklin system expansions converge unconditionally in L1. Finally, it is shown that the Franklin bases characterize the corresponding anisotropic BMO spaces. © 2006 WILEY-VCH Verlag GmbH & Co. KGaA.