Expansion of weak reconstruction sequences to approximate Schauder frames for Banach spaces

It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and weak reconstruction sequences which cannot be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand weak reconstruction sequences to approximate Schauder frames.

Controlled g-fusion frame in Hilbert space

Fusion frame is a generalization of frame, which can analyze signals by projecting them onto multidimensional subspaces. Controlled fusion frame as generalization of fusion frame, it can improve the numerical efficiency of iterative algorithms for inverting the fusion frame operators. In this paper, we first introduce the notion of controlled g-fusion frame, discuss several properties of controlled g-fusion Bessel sequence. Then, we present some sufficient conditions and some characterizations of controlled g-fusion frames. Finally, we study the sum of controlled g-fusion frames.

Signal Representations Via SIP p-frames and SIP Bessel Multipliers in Separable Banach Spaces

Digital signals are often modeled as functions in Banach spaces, such as the ubiquitous [Formula: see text] spaces. The frame theory in Banach spaces induces flexible representations of signals due to the robustness and redundancy of frames. Nevertheless, the lack of inner product in general Banach spaces limits the direct representations of signals in Banach spaces under a given basis or frame. In this paper, we introduce the concept of semi-inner product (SIP) [Formula: see text]-Bessel multipliers to extend the flexibility of signal representations in separable Banach spaces, where [Formula: see text]. These multipliers are defined as composition of analysis operator of an SIP-I Bessel sequence, a multiplication with a fixed sequence and synthesis operator of an SIP-II Bessel sequence. The basic properties of the SIP [Formula: see text]-Bessel multipliers are investigated. Moreover, as special cases, characterizations of [Formula: see text]-Riesz bases related to signal representations are given, and the multipliers for [Formula: see text]-Riesz bases are discussed. We show that SIP [Formula: see text]-Bessel multipliers for [Formula: see text]-Riesz bases are invertible. Finally, the continuity of SIP [Formula: see text]-Bessel multipliers with respect to their parameters is investigated. The results theoretically show that the SIP [Formula: see text]-Bessel multipliers offer a larger range of freedom than frames on signal representations in Banach spaces.

Hilbert–Schmidt frames and their duals

The concept of Hilbert–Schmidt frame (HS-frame) was first introduced by Sadeghi and Arefijamaal in 2012. It is more general than [Formula: see text]-frames, and thus, covers many generalizations of frames. This paper addresses the theory of HS-frames. We present a parametric and algebraic formula for all duals of an arbitrarily given HS-frame; prove that the canonical HS-dual induces a minimal-norm expression of the elements in Hilbert spaces; characterize when an HS-frame is an HS-Riesz basis, and when an HS-Bessel sequence is an HS-Riesz sequence (HS-Riesz basis) in terms of Gram matrices.

On Perturbation of Weighted G−Banach Frames in Banach Spaces

In the present paper, we study perturbation of weighted <em>g</em>−Banach frames in Banach spaces and obtain perturbation results for weighted <em>g</em>−Banach frames. Also, sufficient conditions for the perturbation of weighted <em>g</em>−Banach frames by positively confined sequence of scalars and uniformly scaled version of a given weighted <em>g</em>−Banach Bessel sequence have been given. Finally, we give a condition under which the sum of finite number of sequences of operators is a weighted <em>g</em>−Banach frame by comparing each of the sequences with another system of weighted <em>g</em>−Banach frames in Banach spaces.

GENERALIZED BESSEL AND FRAME MEASURES

Considering a finite Borel measure $ \mu $ on $ \mathbb{R}^d $, a pair of conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(\mu) $, we have introduced $ (p,q) $-Bessel and $ (p,q) $-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we have defined the notions of $ q $-Bessel sequence and $ q$-frame in the semi-inner product space $ L^p(\mu) $. Every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a finite measure $ \mu $. We have constructed a large number of examples of finite measures $ \mu $ which admit infinite $ (p,q) $-Bessel measures $ \nu $. We have showed that if $ \nu $ is a $ (p,q) $-Bessel/frame measure for $ \mu $, then $ \nu $ is $ \sigma $-finite and it is not unique. In fact, by using the convolutions of probability measures, one can obtain other $ (p,q) $-Bessel/frame measures for $ \mu $. We have presented a general way of constructing a $ (p,q) $-Bessel/frame measure for a given measure.

Unconditional convergence constants of g-frame expansions

In this paper, we prove that the unconditional constants of the g-frame expansion in a Hilbert space are bounded by [Formula: see text], where [Formula: see text], [Formula: see text] 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.

Characterizations of (near) exact g-frames, g-Riesz bases, and Besselian g-frames

In this paper, we use a new general sequence corresponding to a [Formula: see text]-Bessel sequence [Formula: see text] to characterize that [Formula: see text] are [Formula: see text]-linear independent, [Formula: see text]-complete and a [Formula: see text]-frame. We also use [Formula: see text] and the refinement [Formula: see text] to characterize each other to be (near) exact [Formula: see text]-frames or [Formula: see text]-Riesz bases. Finally, we give several constructions and an equivalent characterization of Besselian [Formula: see text]-frames and near exact [Formula: see text]-frames.

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

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.

K-g-frames and their dual

This paper is devoted to the study of the dual [Formula: see text]-g-Bessel sequences of [Formula: see text]-g-frames. We firstly make use of the g-preframe operators of a g-Bessel sequence to investigate the constructions of [Formula: see text]-g-frames. And then, taking the g-preframe operators into account, we present several necessary and sufficient conditions under which a g-Bessel sequence is the dual [Formula: see text]-g-Bessel sequence of a given [Formula: see text]-g-frame. We also give a simple example to show that the duality of [Formula: see text]-g-frames is not transitive.