The effect of perturbations of frames on their alternate and approximately dual frames

Abstract

Approximately dual frames as a generalization of duality notion in Hilbert spaces have applications in Gabor systems, wavelets, coorbit theory, and sensor modeling. In recent years, the computing of the associated deviations of the canonical and alternate dual frames from the original ones has been considered by functional analysts. In this paper, we consider the quantitative measurement of the associated deviations of the alternate and approximately dual frames from the original ones. More precisely, it is proved that if the sequence Ψ is sufficiently close to the frame Φ, then for given approximately dual frame Φad of Φ, the approximately dual frame Ψad of Ψ can be found which is close to Φad, and particularly, we estimate their deviation in terms of the distance of the operators and and their approximation rates, where TX and UX denote the synthesis and analysis operators of the frame X, respectively. It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.