This research proposes a framework for frame theory in Dirichlet spaces under relaxed heat kernel conditions, investigating weaker upper Gaussian bounds, polynomial decay, and localized estimates. It introduces a new characterization of frames and establishes stability results for frame bounds under perturbations.
Key findings
Novel relaxed frame condition based on upper heat kernel bounds and generalized cutoff energy inequality.
Frame bounds depend continuously on constants in relaxed heat kernel estimates.
Explicit frame constructions for sub-Gaussian settings, including polynomial analogues.
Characterization of approximation spaces associated with relaxed frames, extending Besov spaces theory.
Limitations & open questions
Theoretical framework needs practical application validation in more diverse spaces.
Stability analysis under broader perturbations remains an open question.