On the type of convergence in atomic representations

Abstract

The type of convergence in atomic representations of spaces of Besov and Triebel–Lizorkin is usually presented in the sense of the topology of the tempered distributions, occasionally with some remarks about the possibility of the convergence being valid in some Lebesgue spaces, if some conditions are met. Until now we are not aware of any explicit indication that those representations usually converge in the Besov or Triebel–Lizorkin spaces themselves. Yet this is indeed the case, as explained in this note. We also deal with a corresponding question for wavelet representations in a recently introduced class of generalized local Hardy spaces.

Keywords:

• Besov spaces
• Triebel–Lizorkin spaces
• atomic representations
• Hardy spaces
• wavelet representations
• Schauder bases

AMS Subject Classifications::

• 46E35
• 42C40

Acknowledgements

This research was supported by Fundação para a Ciência e a Tecnologia (Portugal) through Centro de I&D em Matemática e Aplicações of the University of Aveiro.