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

Approximately dual frames as a generalization of duality notion in Hilbert spaces have applications Gabor systems, wavelets, coorbit theory, and sensor modeling. In recent years, the computing associated deviations canonical alternate from original ones has been considered by functional analysts. this paper, we consider quantitative measurement approximately ones. More precisely, it is proved that if sequence Ψ sufficiently close to frame Φ, then for given Φad Ψad can be found which Φad, particularly, estimate their deviation terms distance operators A 1 : = T Φ U d 2 approximation rates, where TX UX denote synthesis analysis X, respectively. It worth mentioning some our perturbation conditions are quite different those used previous literatures on topic.