(p, q)-Frame measures on LCA groups: Perturbations and Construction

Abstract: Motivated to generalize the Fourier frame concept to Banach spaces we introduce (p, q)-Bessel/frame measures for a given finite measure on LCA groups. We also present a general way of constructing (p, q)-Bessel/frame measures for a given measure. Moreover, we prove that if a measure has an associated (p, q)-frame measure, then it must have a certain uniformity in the sense that the weight is distributed quite uniformly on its support. Next, we show that if the measures $\mu$ and $\lambda$ without atoms whose supports form a packing pair, then $\mu\ast\lambda+\delta_g\ast\mu$ does not admit any (p, q)-frame measure. Finally, we analyze the stability of (p, q)-frame measures under small perturbations. We prove new theorems concerning the stability of (p, q)-frame measures under perturbation in both Hilbert spaces and Banach spaces.