Frame Multipliers Research Articles

Gabor frame multipliers and Parseval duals on the half real line

Recently, Gabor analysis on locally compact abelian (LCA) groups has become the focus of an active research. In practice, the time variable cannot be negative. The half real line [Formula: see text] is an LCA group under multiplication and the usual topology, with the Haar measure [Formula: see text]. This paper addresses Gabor frame multipliers and Parseval duals for [Formula: see text]. We introduce and characterize Gabor frame multipliers and Parseval Gabor frame multipliers based on Zak transform matrices. Our Zak transform matrix is essentially different from the conventional Zibulski–Zeevi matrix. It allows us to define Gabor frame generators by designing suitable matrix-valued functions of finite size. We also prove that an arbitrary Gabor frame [Formula: see text] admits a Parseval dual frame/tight dual frame whenever [Formula: see text] are rational numbers not greater than [Formula: see text].

Quantitative bounds for unconditional pairs of frames

We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [22]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that for all C>0 and N∈N the following is true: Let (xj)j=1N and (fj)j=1N be sequences in a finite dimensional Hilbert space which satisfy ‖xj‖=‖fj‖ for all 1≤j≤N and‖∑j=1Nεj〈x,fj〉xj‖≤C‖x‖,for all x∈ℓ2M and |εj|=1. If the frame operator for (fj)j=1N has eigenvalues λ1≥...≥λM and β>0 is such that λ1≤βM−1∑j=1Mλj then (fj)j=1N has Bessel bound 27β2C. The same holds for (xj)j=1N.

Estimate of the spectral radii of Bessel multipliers and consequences

Bessel multipliers are operators defined from two Bessel sequences of elements of a Hilbert space and a complex sequence, and have frame multipliers as particular cases. In this paper an estimate of the spectral radius of a Bessel multiplier is provided involving the cross Gram operator of the two sequences. As an upshot, it is possible to individuate some regions of the complex plane where the spectrum of a multiplier of dual frames is contained.

Annoyance of railway curve squeal

Trains passing through a curve frequently produce the so-called curve squeal, which are salient noise components (typically either tonal or transient) covering a wide frequency range. Although the main underlying acoustical mechanisms are well known, due to the large variety of curve squeal characteristics, their effects on acoustic parameters and in particular on the perception are difficult to quantify. The work presented here aims at investigating the effects of acoustically described curve squeals on the perceived annoyance, tested in the laboratory with 30 listeners. Measurements of curve squeal at distances of 7.5, 25, and 50 m in three narrow curves were obtained. By means of so-called frame multipliers, time-variable salient features were manipulated and combined with a number of clean rolling noises to simulate squeals with parameters obtained from existing emission measurements. The annoyance ratings showed that the perception of curve squeal can be considerably altered compared to that of regular passby noises. A model was developed for the annoyance equivalent level adjustment from a clean rolling noise based on the contrast of high and mid-frequency energy. The model predictions can serve as a basis to establish rating levels predicting the perceived annoyance of railway curve squeal.

Biframes and some of their properties

Recently, frame multipliers, pair frames, and controlled frames have been investigated to improve the numerical efficiency of iterative algorithms for inverting the frame operator and other applications of frames. In this paper, the concept of biframe is introduced for a Hilbert space. A biframe is a pair of sequences in a Hilbert space that applies to an inequality similar to a frame inequality. Also, it can be regarded as a generalization of controlled frames and a special kind of pair frames. The basic properties of biframes are investigated based on the biframe operator. Then, biframes are classified based on the type of their constituent sequences. In particular, biframes for which one of the constituent sequences is an orthonormal basis {e_{k}}_{k=1}^{infty} are studied. Then, a new class of Riesz bases denoted by [{e_{k}}] is introduced and is called b-Riesz bases. An interesting result is also proved, showing that the set of all b-Riesz bases is a proper subset of the set of all Riesz bases. More precisely, b-Riesz bases induce an equivalence relation on [{e_{k}}].

The essence of invertible frame multipliers in scalability

The essence of invertible frame multipliers in scalability

Continuous frames in tensor product Hilbert spaces, localization operators and density operators

Continuous frames and tensor products are important topics in theoretical physics. This paper combines those concepts. We derive fundamental properties of continuous frames for tensor product of Hilbert spaces. This includes, for example, the consistency property, i.e. preservation of the frame property under the tensor product, and the description of the canonical dual tensors by those on the Hilbert space level. We show the full characterization of all dual systems for a given continuous frame, a result interesting by itself, and apply this to dual tensor frames. Furthermore, we discuss the existence on non-simple tensor product (dual) frames. Continuous frame multipliers and their Schatten class properties are considered in the context of tensor products. In particular, we give sufficient conditions for obtaining partial trace multipliers of the same form, which is illustrated with examples related to short-time Fourier transform and wavelet localization operators. As an application, we offer an interpretation of a class of tensor product continuous frame multipliers as density operators for bipartite quantum states, and show how their structure can be restricted to the corresponding partial traces.

Fusion Frame Duals and Its Applications to Invertible Bessel Fusion Multipliers

Two notions of Bessel fusion multipliers have been recently appeared in the literature, but it seems that some questions of invertibility of fusion frame multipliers have not been satisfactorily answered. To achieve our main research goal, first we survey the approaches toward dual fusion frames existing in the literature and agree on the notion of duality for fusion frames in the sense of Kutyniok et al. (Oper Matrices 11:301–336, 2017). We then proposed a slightly modified version of those notions and analyze how our notion of Bessel fusion multipliers performs with respect to a list of desiderata which, to our minds, an invertible fusion frame multiplier should satisfy. Particularly, in contrast to the existing notions, we show that in many cases Bessel fusion multipliers in our sense behave similar to ordinary Bessel multipliers. Also, special attention is devoted to the study of invertible Bessel fusion multipliers.

Duals and multipliers of K-fusion frames

In this paper, we introduce the concept of K-fusion frames and propose the duality for such frames. The relation between the local frames of K-fusion frames with their duals is studied. The elements from the range of a bounded linear operator K can be reconstructed by K-frames. Also, we establish K-fusion frame multipliers and investigate reconstruction of the range of the operator K by them.

Peak detection for MALDI mass spectrometry imaging data using sparse frame multipliers

MALDI mass spectrometry imaging (MALDI MSI) is a spatially resolved analytical tool for biological tissue analysis by measuring mass-to-charge ratios of ionized molecules. With increasing spatial and mass resolution of MALDI MSI data, appropriate data analysis and interpretation is getting more and more challenging. A reliable separation of important peaks from noise (aka peak detection) is a prerequisite for many subsequent processing steps and should be as accurate as possible. We propose a novel peak detection algorithm based on sparse frame multipliers, which can be applied to raw MALDI MSI data without prior preprocessing. The accuracy is evaluated on a simulated data set in comparison with state-of-the-art algorithms. These results also show the proposed method's robustness to baseline and noise effects. In addition, the method is evaluated on real MALDI-TOF data sets, whereby spatial information can be included in the peak picking process. SignificanceThe field of proteomics, in particular MALDI Imaging, encompasses huge amounts of data. The processing and preprocessing of this data in order to segment or classify spatial structures of certain peptides or isotope patterns can hence be cumbersome and includes several independent processing steps. In this work, we propose a simple peak-picking algorithm to quickly analyze large raw MALDI Imaging data sets, which has a better sensitivity than current state-of-the-art algorithms. Further, it is possible to get an overall overview of the entire data set showing the most significant and spatially localized peptide structures and, hence, contributes all data driven evaluation of MALDI Imaging data.

SOME RESULTS OF K-FRAMES AND THEIR MULTIPLIERS

K -frames are strong tools for the reconstruction of elements from range of a bounded linear operator K on a separable Hilbert space H. In this paper, we study some properties of K -frames and introduce the K -frame multipliers. We also focus on representing elements from the range of K by K -frame multipliers.

Modular frames and invertibility of multipliers in Hilbert C*-modules

ABSTRACT We investigate some properties for Hilbert -modular frames. In the present paper, we also investigate some conditions for invertibility of multipliers in Hilbert -module spaces. We show that by appropriate conditions for symbols, the inverse of an invertible frame multiplier can be represented as a frame multiplier. Dual frames play important role in this representation. We prove and generalize some of the results obtained for frames in Hilbert spaces to Hilbert -modules.

Multipliers for von Neumann–Schatten Bessel sequences in separable Banach spaces

In this paper we introduce the concept of von Neumann–Schatten Bessel multipliers in separable Banach spaces and obtain some of their properties. We study their behavior when the symbol belongs to ℓp. Also, we investigate the continuous dependency of Hilbert–Schmidt Bessel multipliers on their parameters. Special attention is devoted to the study of invertible Hilbert–Schmidt frame multipliers which is an extensive class of ordinary frame multipliers and includes g-frame multipliers and fusion frame multipliers as elementary examples. Among other things, the invertibility of Hilbert–Schmidt Riesz multipliers is completely characterized. In particular, we show that all the existing frames in Hilbert spaces is uniquely determined by the set of its dual frames.

Invariances of the Operator Properties of Frame Multipliers Under Perturbations of Frames and Symbol

ABSTRACTLet Φ and Ψ be frames for ℋ and let Mm,Φ,Ψ be a frame multiplier with the symbol m. In this paper, we restrict our investigation to show that the operator properties of Mm,Φ,Ψ are stable under the perturbations of Φ, Ψ, and m. Also, special attention is devoted to the study of invertible frame multipliers.

The unconditional constants for Hilbert space frame expansions

The most fundamental notion in frame theory is the frame expansion of a vector. Although it is well known that these expansions are unconditionally convergent series, no characterizations of the unconditional constant were known. This has made it impossible to get accurate quantitative estimates for problems which require using subsequences of a frame. We will prove some new results in frame theory by showing that the unconditional constants of the frame expansion of a vector in a Hilbert space are bounded by BA, where A,B are the frame bounds of the frame. Tight frames thus have unconditional constant one, which we then generalize by showing that Bessel sequences have frame expansions with unconditional constant one if and only if the sequence is an orthogonal sum of tight frames. We give further results concerning frame expansions, in which we examine when BA is actually attained or not. We end by discussing the connections of this work to frame multipliers. These results hold in both real and complex Hilbert spaces.

The invertibility of fusion frame multipliers

In this paper, we present fusion frame multipliers and study their properties. In fact, by an efficient approach to dual fusion frames, we introduce the fusion frame multipliers as a generalization of (discrete) frame multipliers and extend the results of frame multipliers. However, many properties of fusion frame multipliers are not satisfied in this case.

On the Dual Frame Induced by an Invertible Frame Multiplier

Recently it has been established that given an invertible frame multiplier with semi-normalized symbol, a specific dual of any of the two involved frames can be determined by the inversion process. The inverse can be represented as a multiplier with the reciprocal symbol, this particular dual of one of the given frames, and any dual of the other frame. The specific dual is the only one having this property among all Bessel sequences. In this manuscript we extend the results showing that the specific dual with the above mentioned property is unique among all possible sequences. Furthermore, the symbol is allowed to be not necessarily semi-normalized. Finally we characterize cases when the canonical dual frame and the new specific dual frame coincide.

Characterization of matrix Fourier multipliers for A-dilation Parseval multi-wavelet frames

Let [Formula: see text] be a [Formula: see text] expansive integral matrix with [Formula: see text]. This paper investigates matrix Fourier multipliers for [Formula: see text]-dilation Parseval multi-wavelet frames, which are [Formula: see text] matrices with [Formula: see text] function entries, map [Formula: see text]-dilation Parseval multi-wavelet frames of length [Formula: see text] to [Formula: see text]-dilation Parseval multi-wavelet frames of length [Formula: see text], where [Formula: see text]. We completely characterize all matrix Fourier multipliers for [Formula: see text]-dilation Parseval multi-wavelet frames and construct several numerical examples. As Fourier wavelet frame multiplier, matrix Fourier multipliers can be used to derive new [Formula: see text]-dilation Parseval multi-wavelet frames and can help us better understand the basic properties of frame theory.

Operators on Partial Inner Product Spaces: Towards a Spectral Analysis

Given a LHS (Lattice of Hilbert spaces) $V_J$ and a symmetric operator $A$ in $V_J$, in the sense of partial inner product spaces, we define a generalized resolvent for $A$ and study the corresponding spectral properties. In particular, we examine, with help of the KLMN theorem, the question of generalized eigenvalues associated to points of the continuous (Hilbertian) spectrum. We give some examples, including so-called frame multipliers.

Representation of the inverse of a frame multiplier

Certain mathematical objects appear in a lot of scientific disciplines, like physics, signal processing and, naturally, mathematics. In a general setting they can be described as frame multipliers, consisting of analysis, multiplication by a fixed sequence (called the symbol), and synthesis. In this paper we show a surprising result about the inverse of such operators, if any, as well as new results about a core concept of frame theory, dual frames. We show that for semi-normalized symbols, the inverse of any invertible frame multiplier can always be represented as a frame multiplier with the reciprocal symbol and dual frames of the given ones. Furthermore, one of those dual frames is uniquely determined and the other one can be arbitrarily chosen. We investigate sufficient conditions for the special case, when both dual frames can be chosen to be the canonical duals. In connection to the above, we show that the set of dual frames determines a frame uniquely. Furthermore, for a given frame, the union of all coefficients of its dual frames is dense in ℓ2. We also introduce a class of frames (called pseudo-coherent frames), which includes Gabor frames and coherent frames, and investigate invertible pseudo-coherent frame multipliers, allowing a classification for frame-type operators for these frames. Finally, we give a numerical example for the invertibility of multipliers in the Gabor case.