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

Abstract EN

The concept of canonical dual K-Bessel sequences was recently introduced, a deep study of which is helpful in further developing and enriching the duality theory of K-frames.

In this paper we pay attention to investigating the structure of the canonical dual K-Bessel sequence of a Parseval K-frame and some derived properties.

We present the exact form of the canonical dual K-Bessel sequence of a Parseval K-frame, and a necessary and sufficient condition for a dual K-Bessel sequence of a given Parseval K-frame to be the canonical dual K-Bessel sequence is investigated.

We also give a necessary and sufficient condition for a Parseval K-frame to have a unique dual K-Bessel sequence and equivalently characterize the condition under which the canonical dual K-Bessel sequence of a Parseval K-frame admits a unique dual K⁎-Bessel sequence.

Finally, we obtain a minimal norm property on expansion coefficients of elements in the range of K resulting from the canonical dual K-Bessel sequence of a Parseval K-frame.