Making more approximate oblique dual frame pairs

The concept of approximate oblique dual frame was introduced by Díaz, Heineken and Morillas. It is more general than traditional dual frame, oblique dual frame, and approximate dual frame. This paper addresses constructing more approximate oblique dual frame pairs starting from one given oblique dual frame pair. Using “analysis and synthesis operator”, “portrait”, and “gap” perturbation techniques, we present several sufficient conditions for constructing approximate oblique dual frame pairs under the general Hilbert space setting. As an application, we then focus on constructing approximate oblique dual frame pairs in shift-invariant subspaces of \(L^{2}(\mathbb R)\).