Some Properties of Canonical Dual K-Bessel Sequences for Parseval K-Frames

Abstract

The concept of canonical dual -Bessel sequences was recently introduced, a deep study of which is helpful in further developing and enriching the duality theory of -frames. In this paper we pay attention to investigating the structure of the canonical dual -Bessel sequence of a Parseval -frame and some derived properties. We present the exact form of the canonical dual -Bessel sequence of a Parseval -frame, and a necessary and sufficient condition for a dual -Bessel sequence of a given Parsevalframe to be the canonical dual -Bessel sequence is investigated. We also give a necessary and sufficient condition for a Parseval -frame to have a unique dual -Bessel sequence and equivalently characterize the condition under which the canonical dual -Bessel sequence of a Parseval -frame admits a unique dual * -Bessel sequence. Finally, we obtain a minimal norm property on expansion coefficients of elements in the range of resulting from the canonical dual -Bessel sequence of a Parseval -frame.

Matrix methods for perfect signal recovery underlying range space of operators

“…In the case that $$$ F $$$ is a Parseval $$$ K $$$‐frame Lemma 2 can be reduced to a result of Xiang 28 . The next result presents the canonical $$$ K $$$‐dual of some classes of $$$ K $$$‐frames.…”

“…In the case that F is a Parseval K-frame Lemma 2 can be reduced to a result of Xiang. 28 The next result presents the canonical K-dual of some classes of K-frames. The computations are relatively straightforward, so we provide a sketch of the proof for convenience of the reader.…”