Multipliers for p-Bessel Sequences in Banach Spaces

Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the continuous dependency of the multipliers on their parameters.     

Bessel fusion multipliers

In this paper we study properties of a Bessel multiplier when the symbol involved belongs to lp. Furthermore, we introduce the concept of Bessel fusion multiplier which generalizes a Bessel multiplier for Bessel fusion sequences. We study their behavior when the symbol belongs to lp and some continuity properties.     

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 ?²${?}^2$. 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.     

Calderón–Zygmund Operators in the Bessel Setting for All Possible Type Indices

We show that many harmonic analysis operators in the Bessel setting,including maximal operators,Littlewood–Paley–Stein type square functions,multipliers of Laplace or Laplace–Stieltjes transform type and Riesz transforms are,or can be viewed as,Calderón–Zygmund operators for all possible values of type parameter λ in this context.This extends results existing in the literature,but being justified only for a restricted range of λ.     

On Gauss-Weierstrass semigroups and inversion of potentials associated with the Bessel differential operator

In this paper, we define the generalized Gauss Weierstrass semigroups with Weierstrass kernel, and give some of their properties. Using them, we study the inversion formulas for the generalized Riesz and Bessel potentials, generated by the generalized shift operators and associated with the Laplace Bessel differential operator.     

Banach空间上p-fusion框架的若干等价描述

本文说明Banach空间上p-fusion框架和p-框架有紧密联系.应用分析算子和合成算子给出p-fusion Bessel序列、p-fusion框架和q-fusion Riesz基的等价描述.     

Representation of Operators in the Time-Frequency Domain and Generalized Gabor Multipliers

Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator’s best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator’s best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.     

Limiting weak-type behaviors for Riesz transforms and maximal operators in Bessel setting

We establish the limiting weak type behaviors of Riesz transforms associated to the Bessel operators on IR+. which are closely related to the best constants of the weak type (1,1) estimates for such operators. Meanwhile, the corresponding results for Hardy-Littlewood maximal operator and fractional maximal operator in Bessel setting are also obtained.     

On the Orthogonality of Frames and the Density and Connectivity of Wavelet Frames

We examine some recent results of Bownik on density and connectivity of the wavelet frames. We use orthogonality (strong disjointness) properties of frame and Bessel sequences, and also properties of Bessel multipliers (operators that map wavelet Bessel functions to wavelet Bessel functions). In addition we obtain an asymptotically tight approximation result for wavelet frames.     

Pair frames in Hilbert $$\varvec{C^*}$$-modules

In this paper, we introduce pair frames in Hilbert \(C^*\)-modules and show that they share many useful properties with their corresponding notions in Hilbert spaces. We also obtain the necessary and sufficient conditions for a standard Bessel sequence to construct a pair frame and get the necessary and sufficient conditions for a Hilbert \(C^*\)-module to admit a pair frame with a symbol and two standard Bessel sequences. Moreover by generalizing some of the results obtained for Bessel multipliers in Hilbert \(C^*\)-modules to pair frames and considering the stability of pair frames under invertible operators, we construct new pair frames and show that pair frames are stable under small perturbations.     

Frame expansions for Gabor multipliers

Discrete Gabor multipliers are composed of rank one operators. We shall prove, in the case of rank one projection operators, that the generating operators for such multipliers are either Riesz bases (exact frames) or not frames for their closed linear spans. The same dichotomy conclusion is valid for general rank one operators under mild and natural conditions. This is relevant since discrete Gabor multipliers have an emerging role in communications, radar, and waveform design, where redundant frame decompositions are increasingly applicable.     

Generalized Bessel and Riesz Potentials on Metric Measure Spaces

We introduce generalized Bessel and Riesz potentials on metric measure spaces and the corresponding potential spaces. Estimates of the Bessel and Riesz kernels are given which reflect the intrinsic structure of the spaces. Finally, we state the relationship between Bessel (or Riesz) operators and subordinate semigroups.      

Fourier–Bessel-type series: The fourth-order case

The structured higher-order Bessel-type linear ordinary differential equations were first discovered in 1994. There is a denumerable infinity of these higher-order equations, all of then of even-order.These differential equations possess many of the properties of the classical second-order Bessel differential equation, but these higher-order cases bring remarkable new analytic structures. In many ways it is sufficient to study the properties of the fourth-order Bessel-type differential equation to be able to assess the corresponding properties of the sixth-and higher-order cases.This paper follows a number of earlier papers devoted to the study of the fourth-order case. These publications show the connections between the special function properties of solutions of the differential equation, and the properties of linear differential operators generated by the associated linear differential expression in certain weighted Lebesgue, and Lebesgue–Stieltjes function spaces.To follow the earlier papers on the study of the fourth-order Bessel-type differential equation, this present paper determines the form of the Fourier–Bessel-type series which best extends the classical theory of the second-order Fourier–Bessel series.In fact the Fourier–Bessel-type series are based on a new orthogonal system in terms of the regular eigensolutions of the fourth-order Bessel-type equation. The corresponding eigenvalues are obtained by restricting the spectral parameter to the zeros of an analytic function arising already in the Dini boundary conditions.     

The Fourth-order Bessel–type Differential Equation

The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane.

There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation.

When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coefficients. Embedded in this form of the equation is a spectral parameter; this combination leads to the generation of self-adjoint operators in a weighted Hilbert function space. In the second-order case one of these associated operators has an eigenfunction expansion that leads to the Hankel integral transform.

This article is devoted to a study of the spectral theory of the Bessel-type differential equation of order four; considered on the positive real axis this equation has singularities at both end-points. In the associated Hilbert function space these singular end-points are classified, the minimal and maximal operators are defined and all associated self-adjoint operators are determined, including the Friedrichs self-adjoint operator. The spectral properties of these self-adjoint operators are given in explicit form.

From the properties of the domain of the maximal operator, in the associated Hilbert function space, it is possible to obtain a virial theorem for the fourth-order Bessel-type differential equation.

There are two solutions of this fourth-order equation that can be expressed in terms of classical Bessel functions of order zero and order one. However it appears that additional, independent solutions essentially involve new special functions not yet defined. The spectral properties of the self-adjoint operators suggest that there is an eigenfunction expansion similar to the Hankel transform, but details await a further study of the solutions of the differential equation.     

Maximal operators, Riesz transforms and Littlewood-Paley functions associated with Bessel operators on BMO

In this paper we study boundedness properties of certain harmonic analysis operators (maximal operators for heat and Poisson semigroups, Riesz transforms and Littlewood-Paley g-functions) associated with Bessel operators, on the space BMOo(R) that consists of the odd functions with bounded mean oscillation on R.     

Time-domain characterization of multiwindow Gabor systems on discrete periodic sets

This paper addresses multiwindow Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. We give some necessary and/or sufficient conditions for multiwindow Gabor systems to foe frames on discrete periodic sets, and characterize two multiwindow Gabor Bessel sequences to foe dual frames on discrete periodic sets. For a given multiwindow Gabor frame, we derive all its Gabor duals, among which we obtain an explicit expression of the canonical Gabor dual. In addition, we generalize multiwindow Gabor systems to the case of a different sampling rate for each window, and investigate multiwindow Gabor frames and dual frames in this case. We also show the properties of the multiwindow Gabor systems are essentially not changed by replacing the exponential kernel with other kernels.     

紧致齐性空间上的调和分析（Ⅳ）─—Riesz变换与Bessel变换

本文研究了紧致齐性空间上的Ｒｉｅｓｚ位势算子与Ｂｅｓｓｅｌ位势算子，Ｒｉｅｓｚ变换与Ｂｅｓｓｅｌ变换，给出了上述算子对应的核函数的具体构造并证明了Ｒｉｅｓｚ变换与Ｂｅｓｓｅｌ变换作为奇异积分算子的Ｈ￣ｐ有界性，ｐ＞０。     

Bessel sequences,wavelet frames,duals and extensions

From the perspectives of duality and extensions, Gabor frames and wavelet frames have contrasting behaviour. Our chief concern here is about duality. Canonical duals of wavelet frames may not be wavelet frames, whereas canonical duals of Gabor frames are Gabor frames. Keeping these in view, we give several constructions of wavelet frames with wavelet canonical duals. For this, a simple characterisation of Bessel sequences and a general commutativity result are given, the former also leading naturally to some extension results.     

Finite Interval Convolution Operators with Transmission Property

We study convolution operators in Bessel potential spaces and (fractional) Sobolev spaces over a finite interval. The main purpose of the investigation is to find conditions on the convolution kernel or on a Fourier symbol of these operators under which the solutions inherit higher regularity from the data. We provide conditions which ensure the transmission property for the finite interval convolution operators between Bessel potential spaces and Sobolev spaces. These conditions lead to smoothness preserving properties of operators defined in the above-mentioned spaces where the kernel, cokernel and, therefore, indices do not depend on the order of differentiability. In the case of invertibility of the finite interval convolution operator, a representation of its inverse is presented in terms of the canonical factorization of a related Fourier symbol matrix function.     

Frames of Periodic Shift-Invariant Spaces

This note studies Bessel sequences and frames of shift-invariant spaces generated by a countable set of periodic functions. We give characterizations under which the set of translations of the countable set is a Bessel sequence or a frame in terms of spectral decompositions of some self-adjoint operators.