2019/04/30 by Mahin Hajiabootorabi, Hajiabootorabi, M., Hossein Javanshiri +5
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Optical measurement and interference techniques #Photoacoustic and Ultrasonic Imaging #Ultrasonics and Acoustic Wave Propagation
paper · pdf · doi:10.48550/arxiv.1905.00113
openalex publication_date 2019/04/30 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
Approximately dual frames as a generalization of duality notion in Hilbert\nspaces have applications in Gabor systems, wavelets, coorbit theory and sensor\nmodeling. In recent years, the computing of the associated deviations of the\ncanonical and alternate dual frames from the original ones has been considered\nby some authors. However, the quantitative measurement of the associated\ndeviations of the alternate and approximately dual frames from the original\nones has not been satisfactorily answered. In this paper, among other things,\nit is proved that if the sequence \Ψ=(\ψn)n is sufficiently close to\nthe frame \Φ=(\φn)n, then \Ψ is a frame for mathcal H and\napproximately dual frames \Φad=(\φadn)n and\n\Ψad=(\ψadn)n can be found which are close to each other and\nparticularly, we estimate the deviation from perfect reconstruction in terms of\nthe operator mathcal A1:=T_\Φ U\Φad and mathcal A2:=T_\Ψ\nU\Ψad and their approximation rates, where TX and UX denote the\nsynthesis and analysis operators of the frame X, respectively. Finally, we\ndemonstrate how our results apply in the practical case of Gabor systems. It is\nworth mentioning that some of our perturbation conditions are quite different\nfrom those used in the previous literatures on this topic.\n