On a class of weak R-duals and the duality relations (Q2175552)

In this paper, the authors introduce and study weak R-dual sequences for given sequences in a Hilbert space with respect to two Parseval frames. Weak R-dual sequences are R-dual sequences with relaxation of replacing orthonormal bases with Parseval frames. A necessary and sufficient condition for Bessel sequences to be frames is given in terms of weak R-dual sequences. Few results concerning equivalence of two sequences are proved in terms of the relationship between their weak R-dual sequences. In the last section of the paper, the authors provide an expression of the canonical duals of weak R-dual sequences.