Some properties of generalized and approximately dual frames in Hilbert spaces

2015/09/25 by Hossein Javanshiri, Javanshiri, Hossein
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems #Primary: 42C15 #Secondary: 47A58

paper · pdf · doi:10.48550/arxiv.1509.07812

openalex publication_date 2015/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the present paper, some sufficient and necessary conditions for two frames Φ=(φ_n)_n and Ψ=(ψ_n)_n under which they are approximately or generalized dual frames are determined depending on the properties of their analysis and synthesis operators. We also give a new characterization for approximately dual frames associated with a given frame and given operator by using of bounded operators. Among other things, we prove that if two frames Φ=(φ_n)_n and Ψ=(ψ_n)_n are close to each other, then we can find approximately dual frames Φ^ad=(φ^ad_n)_n and Ψ^ad=(ψ^ad_n)_n of them which are close to each other and T_ΦU_Φ^ad=T_ΨU_Ψ^ad, where T_Φ and T_Ψ (resp. U_Φ^ad and U_Ψ^ad) are the analysis operators (resp. synthesis operators) of the frames Φ and Ψ (resp. Φ^ad and Ψ^ad), respectively. We then give some consequences on generalized dual frames. Finally, we apply these results to find some construction results for approximately dual frames for a given Gabor frame.

Related

• Approximately dual frame pairs in Hilbert spaces and applications to Gabor frames
• The effect of perturbations of frames on their alternate and\n approximately dual frames
• K-g-frames and approximate K-g-duals in Hilbert spaces
• Dual pair and Approximate dual for continuous frames in Hilbert spaces
• Some Results about Frames