Perturbation and construction of almost self‐located robust frames with applications to erasure recovery

Abstract

In the frame theory, stability is one of the most important properties of frames. In this paper, we investigate the stability of the almost self-located robust frames, which can reconstruct signals from its unordered partial frame coefficients. We give conditions on the stability of almost self-located robustness of frames in erasure recovery. We prove that the property of almost self-located robustness is preserved with respect to an invertible operator, and consequently, every almost self-robust frame can be transformed into an almost self-located robust Parseval frame by using a special invertible transformation. We also present some construction methods of almost self-located robust frames. In particular, we show that every frame can be rescaled to an almost self-located frame after adding a suitable vector.

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

“…27 Although these questions give some sufficient conditions under which perturbations preserve Hilbert frames, but also they are of utmost importance from different point of view. For example, a perturbation question has been used in Jiang et al 28 for obtaining new almost self-located robust frames from the given one, and Christensen and Laugesen 6, Theorems 4.3 and 4.5 used perturbation theory for obtaining an almost perfect reconstruction formula in Hilbert spaces. In this section, motivated by Christensen and Laugesen 6 and Kutyniok et al, 11 we study how perturbation effects the approximately dual frames of original and perturbed sequence and estimate the error in the approximation of functions belonging to the perturbed space.…”