Volume 30, Issue 4

We consider the Hankel multidimensional operator Δ α \Delta_{\alpha} , α = ( α 1 , … , α n ) ∈ ] − 1 2 , + ∞ [ n \alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathopen{]}-\frac{1}{2},+\infty\mathclose{[}^{n} , defined on ] 0 , + ∞ [ n \mathopen{]}0,+\infty\mathclose{[}^{n} by Δ α = ∑ j = 1 n ( ∂ 2 ∂ x j 2 + 2 ⁢ α j + 1 x j ⁢ ∂ ∂ x j ) . \Delta_{\alpha}=\sum_{j=1}^{n}\biggl{(}\frac{\partial^{2}}{\partial x_{j}^{2}}+\frac{2\alpha_{j}+1}{x_{j}}\frac{\partial}{\partial x_{j}}\biggr{)}. We define and study the Wigner transformation V g \mathscr{V}_{g} (called also Gabor transform), where g ∈ L 2 ⁢ ( d ⁢ μ α ) g\in L^{2}(d\mu_{\alpha}) and d ⁢ μ α d\mu_{\alpha} is the measure defined on [ 0 , + ∞ [ n [0,+\infty\mathclose{[}^{n} by d ⁢ μ α ⁢ ( x ) = ⨂ j = 1 n x j 2 ⁢ α j + 1 2 α j ⁢ Γ ⁢ ( α j + 1 ) ⁢ d ⁢ x j . d\mu_{\alpha}(x)=\bigotimes_{j=1}^{n}\frac{x_{j}^{2\alpha_{j}+1}}{2^{\alpha_{j}}\Gamma(\alpha_{j}+1)}\,dx_{j}. Using harmonic analysis related to the Hankel operator Δ α \Delta_{\alpha} , we prove a Plancherel theorem and an orthogonality property for the transformation V g \mathscr{V}_{g} . Next, we establish a reconstruction formula for V g \mathscr{V}_{g} and give some applications. In the second part of this work, as applications of the Wigner transformation V g \mathscr{V}_{g} , we define and study the anti-Wick operators A g 1 , g 2 ⁢ ( σ ) \mathscr{A}_{g_{1},g_{2}}(\sigma) , where g 1 , g 2 ∈ L 2 ⁢ ( d ⁢ μ α ) g_{1},g_{2}\in L^{2}(d\mu_{\alpha}) are called window functions and σ ∈ L p ⁢ ( d ⁢ μ α ⊗ d ⁢ μ α ) \sigma\in L^{p}(d\mu_{\alpha}\otimes d\mu_{\alpha}) is a signal. Building on the properties of the Wigner transformation V g \mathscr{V}_{g} , we prove that the operators A g 1 , g 2 ⁢ ( σ ) \mathscr{A}_{g_{1},g_{2}}(\sigma) are bounded linear operators and compact on the Hilbert space L 2 ⁢ ( d ⁢ μ α ) L^{2}(d\mu_{\alpha}) . Finally, we establish a formula of the trace for the anti-Wick operator A g 1 , g 2 ⁢ ( σ ) \mathscr{A}_{g_{1},g_{2}}(\sigma) when the signal 𝜎 belongs to L 1 ⁢ ( d ⁢ μ α ⊗ d ⁢ μ α ) L^{1}(d\mu_{\alpha}\otimes d\mu_{\alpha}) .

In this paper, we develop some generalized Fredholm perturbation results for bounded linear operators acting on non-reflexive Banach spaces satisfying certain conditions. Furthermore, we investigate the stability of the generalized Schechter S -essential spectrum by means of the so-called generalized Fredholm index. This concept is introduced as a semigroup homomorphism satisfying certain properties. Some generalized Fredholm results for block operator matrices acting on a non-reflexive product of two Banach spaces are also developed.

We show that the “one-radius” spherical mean value property is sufficient to characterize harmonic tempered distributions subject to the classical Laplace operator.

Stefan Banach proved that even the Fourier series of the function f ⁢ ( x ) = 1 {f(x)=1} ( x ∈ [ 0 , 1 ] ) {(x\in[0,1])} might not be convergent for some orthonormal systems (ONS). Thus we can conclude that the Fourier series of functions belonging to a certain differential class cannot be convergent in general. On the other hand, for classical ONS (trigonometric, Haar, Walsh systems, …), the convergence of Fourier series of differentiable class functions is a simple subject. In this paper, we investigate the sequence of positive numbers such that, multiplying the Fourier coefficients of Lipschitz class functions by these numbers, one obtains a convergent series of the special form. From the convergence of these special form series, we derive the convergence of the special Fourier series of Lipschitz class functions with respect to general ONS. The obtained results are best possible.

In this paper, we introduce the notions of natural transformation between morphisms of quasigroupoids and between morphisms of weak Hopf quasigroups. Also, we prove that natural transformations between morphisms of finite quasigroupoids come from natural transformations between morphisms of weak Hopf quasigroups and, on the other hand, we obtain that every natural transformation for morphisms defined between pointed cosemisimple weak Hopf quasigroups comes from a natural transformation between finite quasigroupoids.

The nonlinear diffusion system with source terms is studied. This model is derived via Maxwell equations describing electromagnetic field propagation in media. The uniqueness of the solution and its behavior as t → ∞ t\to\infty are proved. We also prove that the finite difference approximation converges in space and time. Numerical implementation with various experiments for different values of the input parameters is performed to validate the theoretical conclusions.

In this work, the subject of ideal in a semiprime ring with multiplicative (generalized)- reverse derivations studied is included. We give new essential results for researchers in this field and generalize some results found in the literature. Also, the application of continuous reverse derivations in Banach algebras is discussed for the first time.

In this paper, we study a boundary value problem for a second-order impulsive p -Laplacian dynamic equation on a time scale. The approach is based on a critical point theorem. Finally, some examples are given to illustrate the results.

In this paper, we investigate the structure of finite solvable groups G admitting a normal subgroup N such that G / N {G/N} is non-abelian and N contains all but four conjugacy classes of G .

The main objective of this paper is to establish multiresolution analysis associated with quadratic-phase Fourier wavelet transform. Moreover, we develop the theory of orthonormal wavelets in the quadratic-phase Fourier transform domain and investigate some of its properties.

In this paper we introduce a weighted Hardy space ℋ β {\mathscr{H}_{\beta}} . This space generalizes some complex Hilbert spaces like the Dirichlet space 𝒟 {\mathscr{D}} , the Bergman space 𝒜 {\mathscr{A}} and the Segal–Bargmann space ℱ {\mathscr{F}} . It plays the role of background for our contribution. In particular, we study the derivative operator D and its adjoint operator L β {L_{\beta}} on ℋ β {\mathscr{H}_{\beta}} . Furthermore, we establish a general uncertainty inequality of Heisenberg type for the space ℋ β {\mathscr{H}_{\beta}} .

Let 𝔊 {\mathfrak{G}} be the Galilean Lie algebra in 2+1 space-times. It is known that there is no faithful representation of 𝔊 {\mathfrak{G}} given by derivations with homogeneous coefficients of degree 1 on the polynomial ring ℝ ⁢ [ x 1 , … , x n ] {\mathbb{R}[x_{1},\dots,x_{n}]} for n ≤ 3 {n\leq 3} . In this paper, we consider such representations of 𝔊 {\mathfrak{G}} for n = 4 {n=4} . By classifying all collections of Galilean matrices of order 4 up to conjugation, which yields a complete classification of faithful 𝔊 {\mathfrak{G}} -modules of dimension 4, we show that all such faithful representations of 𝔊 {\mathfrak{G}} on ℝ ⁢ [ x 1 , … , x 4 ] {\mathbb{R}[x_{1},\dots,x_{4}]} are classified into two types up to equivalence, each of which is parameterized by ℝ 3 {\mathbb{R}^{3}} . As a byproduct, we show that all faithful 𝔊 {\mathfrak{G}} -modules of dimension 4 are indecomposable.

A family of Zygmund-type spaces, called Dirichlet–Zygmund-type spaces, are introduced. The boundedness, compactness and the essential norm of weighted composition operators from Dirichlet–Zygmund-type spaces into Stević-type spaces are also investigated in this paper.