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WEIGHTED AND CONTROLLED FRAMES: MUTUAL RELATIONSHIP AND FIRST NUMERICAL PROPERTIES

Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, A-1040 Vienna, Austria

Corresponding author.

Institut de Physique Théorique, Université Catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium

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ANNA GRYBOŚ

NuHAG, Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090, Austria

Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland

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https://doi.org/10.1142/S0219691310003377Cited by:61 (Source: Crossref)

Abstract

Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we develop systematically these notions, including their mutual relationship. We will show that controlled frames are equivalent to standard frames and so this concept gives a generalized way to check the frame condition, while offering a numerical advantage in the sense of preconditioning. Next, we investigate weighted frames, in particular their relation to controlled frames. We consider the special case of semi-normalized weights, where the concepts of weighted frames and standard frames are interchangeable. We also make the connection with frame multipliers. Finally, we analyze weighted frames numerically. First, we investigate three possibilities for finding weights in order to tighten a given frame, i.e. decrease the frame bound ratio. Then, we examine Gabor frames and how well the canonical dual of a weighted frame is approximated by the inversely weighted dual frame.

Keywords:

AMSC: 42C15, 42C40, 65T60

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