The main focus of this paper is to define the Wigner transform on Chébli-Trimèche hypergroups of exponential growth and to present several related results. Next, we introduce a new class of pseudo-differential operators ℒψ1, ψ2(σ), called localization operators, which depend on a symbol σ and two admissible functions ψ1 and ψ2. We provide criteria, in terms of the symbol σ, for their boundedness and compactness. We also show that these operators belong to the Schatten–von Neumann class Sp for all p ∈ [1, +∞], and we derive a trace formula.
A Young subgroup of the symmetric group \(\mathcal{S}_{N}\) with three factors, is realized as the stabilizer \(G_{n}\) of a monomial \(x^{\lambda}\) ( \(=x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2} }\cdots x_{N}^{\lambda_{N}}\)) with \(\lambda=\left( d_{1}^{n_{1}},d_{2}^{n_{2}},d_{3}^{n_{3}}\right)\) (meaning \(d_{j}\) is repeated \(n_{j}\) times, \(1\leq j\leq3\)), thus is isomorphic to the direct product \(\mathcal{S}_{n_{1}}\times\mathcal{S}_{n_{2}}\times\mathcal{S}_{n_{3}}\). The orbit of \(x^{\lambda}\) under the action of \(\mathcal{S}_{N}\) (by permutation of coordinates) spans a module \(V_{\lambda}\), the representation induced from the identity representation of \(G_{n}\). The space \(V_{\lambda}\) decomposes into a direct sum of irreducible \(\mathcal{S}_{N}\)-modules. The spherical function is defined for each of these, it is the character of the module averaged over the group \(G_{n}\). This paper concerns the value of certain spherical functions evaluated at a cycle which has no more than one entry in each of the three intervals \(I_{j}=\left\{ i:\lambda_{i}=d_{j}\right\} ,1\leq j\leq3\). These values appear in the study of eigenvalues of the Heckman-Polychronakos operators in the paper by V. Gorin and the author (arXiv:2412:01938v1). The present paper determines the spherical function values for \(\mathcal{S}_{N}\)-modules \(V\) of two-row tableau type, corresponding to Young tableaux of shape \(\left[ N-k,k\right]\). The method is based on analyzing the effect of a cycle on \(G_{n}\)-invariant elements of \(V\). These are constructed in terms of Hahn polynomials in two variables.
Hardy-Hilbert-type integral inequalities are among the classics of mathematical analysis. In particular, this includes well-known variants involving homogeneous power-max kernel functions. In this article, we extend the theory by studying the non-homogeneous case using specially designed power-max kernel functions. Additionally, we explore different integration domains to increase the flexibility of our results in a variety of mathematical contexts. We also establish several equivalences, modifications and generalizations of our main integral inequalities. The proofs are detailed and self-contained. To support the theory, we provide numerical examples together with the corresponding implementation codes for transparency and reproducibility.
In this paper, we extend the one-dimensional Gabor transform discussed to the Weinstein harmonic analysis setting. We obtain the expected properties of extended Gabor transform such as inversion formula and Calderon’s reproducing formula.
This paper shows how a family of function spaces, coined as Assiamoua spaces, plays a fundamental role in the Fourier analysis of vector-valued functions on compact groups. These spaces make it possible to transcribe the classical results of Fourier analysis in the framework of analysis of vector-valued functions and vector measures. The construction of Sobolev spaces of vector-valued functions on compact groups rests heavily on the members of the aforementioned family.
This work introduces a unique family of bi-univalent functions utilising \(q\)-Gegenbauer polynomials. The estimates of the initial coefficients \(\left\vert a_{2}\right\vert\) and \(\left\vert a_{2}\right\vert\) for functions in this new class, together with the Fekete-Szegö functional, have been obtained. Subsequent to the specialisation of the parameters utilised in our principal findings, many novel outcomes are presented.
In this paper, we establish some new characterizations of a weight \(w\) such that discrete Hardy operator \(\mathcal{H}f(n):=\frac{1}{n}\sum\limits_{s=1}^{n}f(s)\) for quasi-nonincreasing sequence \(f(n)\) is bounded in the Banach space \(\ell _{w}^{p}(\mathbb{Z}_{+})\) when \(0<p<\infty .\) In particular, we will prove that \(\mathcal{H}f\) is bounded in \(\ell _{w}^{p}(\mathbb{Z}_{+})\) if and only if \(w\) belongs to the \(\beta\)-discrete Arino and Muckenhoupt class \(\mathcal{B}_{p,\beta }\). We prove that the self-improving property for the class \(\mathcal{B}_{p,\beta }\) holds, that is we prove that if \(w\in \mathcal{B}_{p,\beta }\) then there exists an \(\varepsilon >0\) such that \(w\in \mathcal{B}_{p-\varepsilon ,\beta }.\)
In this article, we establish new integral inequalities involving sub-multiplicative functions. We first derive several inequalities of primitive type, followed by new inequalities of the convolution product type. We also obtain integral bounds for functions evaluated on the product of two variables. Finally, we study double integral inequalities and their variations. Simple examples are used to illustrate the theory. The understanding of integral inequalities under submultiplicative assumptions is thus deepened, and some new ideas for further research in mathematical analysis are provided.
The current study focuses on the investigation and develop of a new approach called Hussein–Jassim method (HJM), suggested lately by Hassan et al.; specifically, we investigate its applicability to fractional ordinary delay differential equations in the Caputo fractional sense. Several examples are offered to demonstrate the method’s reliability. The results of this study demonstrate that the proposed method is highly effective and convenient for solving fractional delay differential equations.
We define and study the Stockwell transform \(\mathscr{S}_g\) associated with the Whittaker operator
\[\Delta_{\alpha}:=-\frac{1}{4}\left[x^2\frac{\mbox{d}^2}{\mbox{d}x^2}+(x^{-1}+(3-4\alpha)x)\frac{\mbox{d}}{\mbox{d}x}\right],\]
and prove a Plancherel theorem. Moreover, we define the localization operators \(L_{g,\xi}\) associated to this transform. We study the boundedness and compactness of these operators and establish a trace formula. Finally, we give a Shapiro-type uncertainty inequality for the modified Whittaker-Stockwell transform \(\mathscr{S}_g\).
