Narrow Results

In this paper we present a family of analysis and synthesis systems of operators with frame-like properties for the range of a bounded operator on a separable Hilbert space. This family of operators is called a Θ–g-frame, where Θ is a bounded operator on a Hilbert space. Θ–g-frames are a generalization of g-frames, which allows to reconstruct elements from the range of Θ. In general, range of Θ is not a closed subspace. We also construct new Θ–g-frames by considering Θ–g-frames for its components. We further study Riesz decompositions for Hilbert spaces, which are a generalization of the notion of Riesz bases. We define the coefficient operators of a Riesz decomposition and we will show that these coefficient operators are continuous projections. We obtain some results about stability of Riesz decompositions under small perturbations.

In this paper, we first introduce the notion of controlled weaving $K$-$g$-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving $K$-$g$-frames in separable Hilbert spaces. Also, a characterization of controlled weaving $K$-$g$-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled $K$-$g$-woven frames give ordinary weaving $K$-frames and vice-versa.

The results in this paper can be divided into three parts. First, we generalize the recent results of Benavente, Christensen and Zakowicz on approximately dual generalized shift-invariant frames on the real line to generalized translation-invariant (GTI) systems on locally compact abelian (LCA) groups. Second, we explain in detail how GTI frames can be realized as $g$-frames. Finally, the known results on perturbation of $g$-frames and results on perturbation of generalized shift-invariant systems are applied and extended to GTI systems on LCA groups.

In this paper, we give some new characterizations of g-frames, g-Bessel sequences and g-Riesz bases from their topological properties. By using the Gram matrix associated with the g-Bessel sequence, we present a sufficient and necessary condition under which the sequence is a g-Bessel sequence (or g-Riesz basis). Finally, we consider the excess of a g-frame and obtain some new results.

In this paper, we prove that the unconditional constants of the g-frame expansion in a Hilbert space are bounded by $\sqrt{B/A}$, where $A$, $B$ are the frame bounds of the g-frames. It follows that tight g-frames have unconditional constant one. Then we generalize this to a classification of such g-frames by showing that a g-Bessel sequence has unconditional constant one if it is an orthogonal sum of g-tight frames. We also obtain a new result under which a g-Bessel sequence is a g-frame from the view of unconditional constant. Finally, we prove similar results for cross g-frame expansions as long as the cross g-frame expansions stay uniformly bounded away from zero.