Some new results on the weaving of K-g-frames in Hilbert spaces

In this paper, we provide some conditions for a K-woven pair of K-g-frames to be preserved under an operator and particularly, we report that applying two different operators to a K-woven pair of K-g-frames can leave them K-woven. Several new methods on the construction of K-woven pair of K-g-frames are also given. We end the paper with a new perturbation result on the weaving of K-g-frames, which shows that, under the perturbation condition involved in one known result on this topic, two K-g-frames can be K-woven in the whole space, not merely in the subspace Range(K).

Analytic core and quasi-nilpotent part of linear relations in Banach spaces

In this paper, we investigate the notion of analytic core and quasi-nilpotent part of a linear relation. Furthermore, we are interested in studying the set of Generalized Kato linear relations to give some of their properties in connection with the analytic core and the quasi-nilpotent part. We finish by giving a perturbation result for this set of linear relations.

Frames for operators in Banach spaces via semi-inner products

In this paper, the concept of a family of local atoms in a Banach space is introduced by using a semi-inner product (s.i.p.). Then this concept is generalized to an atomic system for operators in Banach spaces. We also give some characterizations of atomic systems leading to new frames for operators. In addition, a reconstruction formula is obtained. The characterizations of atomic systems allow us to state some results for sampling theory in s.i.p reproducing kernel Banach spaces. Finally, we define the concept of frame operator for these kinds of frames in Banach spaces and then we establish a perturbation result in this framework.

Bessel Sequences and Its F-Scalability

Frame theory, which contains wavelet analysis and Gabor analysis, has become a powerful tool for many applications of mathematics, engineering and quantum mechanics. The study of extension principles of Bessel sequences to frames is important in frame theory. This paper studies transformations on Bessel sequences to generate frames and Riesz bases in terms of operators and scalability. Some characterizations of operators that mapping Bessel sequences to frames and Riesz bases are given. We introduce the definitions of F-scalable and P-scalable Bessel sequences. F-scalability and P-scalability of Bessel sequences are discussed in this paper, then characterizations of scalings of F-scalable or P-scalable Bessel sequences are established. Finally, a perturbation result on F-scalable Bessel sequences is derived.

Nonstationary Gabor frames — Existence and construction

Nonstationary Gabor frames were recently introduced in adaptive signal analysis. They represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. In this paper, we show a general existence result for this family of frames. Then, we give a perturbation result for nonstationary Gabor frames and construct nonstationary Gabor frames with non-compactly supported windows from a related painless nonorthogonal expansion. Finally, the theoretical results are illustrated by two examples of practical relevance.