Narrow Results

This paper addresses the weaving theory of operator-valued frames (OPV-frames). We give a rigorous proof of the equivalence between “weakly woven” and “woven” of OPV-frames; estimate the optimal universal OPV-frame bounds of all weavings; and prove that OPV-frame and its dual OPV-frame are woven. Also, using examples, we show that “woven property” does not have transmissibility, and that a collection of pairwise weaving frames need not be woven. Finally, we give a sufficient condition for a collection of adjacent weaving OPV-frames to be woven.

A necessary and sufficient condition for the perturbation of a Banach frame by a non-zero functional to be a Banach frame has been obtained. Also a sufficient condition for the perturbation of a Banach frame by a sequence in E* to be a Banach frame has been given. Finally, a necessary condition for the perturbation of a Banach frame by a finite linear combination of linearly independent functionals in E* to be a Banach frame has been given.

The notion of frame has some generalizations such as frames of subspaces, fusion frames and g-frames. In this paper, we introduce fusion frames and g-frames in Hilbert C*-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. We also generalize a perturbation result in frame theory to g-frames in Hilbert spaces. We also show that tensor product of fusion frames (g-frames) is a fusion frame (g-frame) and tensor product of resolution of identity is a resolution of identity.

This paper deals with the theory of frames for vector-valued Weyl–Heisenberg wavelets (VVWHW). We derive frame and the corresponding frame bounds for VVWHW.

This paper develops several aspects of shift-invariant spaces on locally compact abelian groups. For a second countable locally compact abelian group G we prove a useful Hilbert space isomorphism, introduce range functions and give a characterization of shift-invariant subspaces of L²(G) in terms of range functions. Utilizing these functions, we generalize characterizations of frames and Riesz bases generated by shifts of a countable set of generators from L²(ℝⁿ) to L²(G).

Frames of subspaces for Banach spaces have been introduced and studied. Examples and counter-examples to distinguish various types of frames of subspaces have been given. It has been proved that if a Banach space has a Banach frame, then it also has a frame of subspaces. Also, a necessary and sufficient condition for a sequence of projections, associated with a frame of subspaces, to be unique has been given. Finally, we consider complete frame of subspaces and prove that every weakly compactly generated Banach space has a complete frame of subspaces.

Bi-Banach frames in Banach spaces have been defined and studied. A necessary and sufficient condition under which a Banach space has a Bi-Banach frame has been given. Finally, Pseudo exact retro Banach frames have been defined and studied.

Various types of Schauder frames have been defined and studied. A necessary and sufficient condition for each type of Schauder frame is given. Finally, we give some theoretical applications of these types of Schauder frames.

Casazza and Kutyniok [Frames of subspaces, in Wavelets, Frames and Operator Theory, Contemporary Mathematics, Vol. 345 (American Mathematical Society, Providence, RI, 2004), pp. 87–113] defined fusion frames in Hilbert spaces to split a large frame system into a set of (overlapping) much smaller systems and being able to process the data effectively locally within each sub-system. In this paper, we handle this problem using block sequences and generalized block sequences with respect to g-frames. Examples have been given to show their existence. A necessary and sufficient condition for a block sequence with respect to a g-frame to be a g-frame has been given. Finally, a sufficient condition for a generalized block sequence with respect to a g-frame to be a g-frame has been given.

In this paper, we introduce controlled frames in Hilbert $C^{\ast }$-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. Next, we give a characterization of controlled frames in Hilbert $C^{\ast }$-module. Also multiplier operators for controlled frames in Hilbert $C^{\ast }$-modules will be defined and some of its properties will be shown. Finally, we investigate weighted frames in Hilbert $C^{\ast }$-modules and verify their relations to controlled frames and multiplier operators.

Fusion frames or frames of subspaces are an extension of frames that have many applications in science. The paper presents non-orthogonal fusion frames with real valued bounds and $C^{\ast }$-valued bounds in Hilbert $C^{\ast }$-modules which they are called NOFF and NO${}^{\ast }$FF, respectively. Their elementary properties are studied, for example, the relations between NOFF and NO${}^{\ast }$FF in Hilbert $C^{\ast }$-modules are established. More precisely, we investigate these fusion frames in Hilbert $C^{\ast }$-modules over different $C^{\ast }$-algebras.

A WH-packet is a system of vectors which is analogous to Aldroubi’s model for explicit expression of vectors (including frame vectors) in terms of a series associated with a given frame. In this paper, we study frame properties of WH-packet type system for matrix-valued wave packet frames in the function space $L^2({ℝ}^d,{ℂ}^{s\times r})$. A necessary and sufficient condition for WH-packets of matrix-valued wave packet frames in terms of a bounded below operator is given. We present sufficient conditions for both lower and upper frame conditions on scalars associated with WH-packet of matrix-valued wave packet frames. Finally, a Paley–Wiener type perturbation theorem for WH-packet of matrix-valued wave packet frames is given. Several examples are given to illustrate the results.

In this paper, we introduce and characterize controlled dual frames in Hilbert spaces. We also investigate the relation between bounds of controlled frames and their related frames. Then, we define the concept of approximate duality for controlled frames in Hilbert spaces. Next, we introduce multiplier operators of controlled frames in Hilbert spaces and investigate some of their properties. Finally, we show that the inverse of a controlled multiplier operator is also a controlled multiplier operator under some mild conditions.

Let $L^2(ℝ,ℍ)$ be a space of square integrable quaternionic-valued functions defined on real line. In this paper, for $\mathrm{\Phi }\in L^2(ℝ,ℍ),$$a>1$ and $q>0$, if the sequence of functions $\{a^{m/2}\mathrm{\Phi }(a^{m}x-nq),m,n\in \mathbb{Z}\}$ is a wavelet frame of $L^2(ℝ,ℍ)$, we study the stability of the wavelet frame when the sampling $\{n\}$ or the mother wavelet $\mathrm{\Phi }$ has perturbation by using the theory of wavelet analysis.

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.

Since the introduction of R-duals by Casazza, Kutyniok and Lammers with the motivation to obtain a general version of duality principle in Gabor analysis, various R-duals and some relaxations of the R-dual setup have been introduced and studied by some mathematicians. They provide a powerful tool in the analysis of duality relations in general frame theory, and are far beyond the duality principle in Gabor analysis. In this paper, we introduce the concept of generalized weak R-dual based on a pair of frames which is a relaxation of the R-dual setup. Using generalized weak R-duals, we characterize the frame properties of a sequence and the equivalence between two frames, prove that the generalized weak R-duals of frames (Riesz bases) are frame sequences (frames), and present a coefficient expression corresponding to the canonical duals of generalized weak R-duals. Some examples are provided to illustrate the generality of the theory.

The concept of Hilbert–Schmidt frame (HS-frame) was first introduced by Sadeghi and Arefijamaal in 2012. It is more general than $g$-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.

The theory of Gabor frames has been extensively investigated. This paper addresses partial Gabor systems. We introduce the concepts of partial Gabor system, frame and dual frame. We present some conditions for a partial Gabor system to be a partial Gabor frame, and using these conditions, we characterize partial dual frames. We also give some examples. It is noteworthy that the density theorem does not hold for general partial Gabor systems.

Hilbert–Schmidt frame (HS-frame) has interested some mathematicians in recent years, which is more general than g-frame. This paper addresses near Riesz and Besselian properties of HS-operator sequences. We characterize HS-frame and Riesz properties of g-operator sequences using their DSI-sequences; prove that an HS-Riesz basis is an exact HS-frame while the converse is not true, and an arbitrary HS-Riesz frame contains an HS-Riesz basis; and present the connection among near HS-Riesz property, Besselian property and the kernel space dimension of synthesis operator of an HS-operator sequence.

Due to ${\mathrm{ℝ}}_{+}$ not being a group under addition, $L^2({\mathrm{ℝ}}_{+})$ admits no traditional wavelet or Gabor frame. This paper addresses a class of dilation-and-modulation (${ℳ}{𝒟}$) Parseval frames for $L^2({\mathrm{ℝ}}_{+})$. We present a parametric expression of all ${ℳ}{𝒟}$-Parseval frames (frame sequences) and ${ℳ}{𝒟}$-orthonormal bases (systems); and develop an approach to obtain all ${ℳ}{𝒟}$-Parseval frames (frame sequences) and ${ℳ}{𝒟}$-orthonormal bases (systems) starting with one ${ℳ}{𝒟}$-Parseval frame for $L^2({\mathrm{ℝ}}_{+})$.