arXiv subjects

Trinh Tuan

Publications and source records attributed to Trinh Tuan.

9 recordsLinked to original sources

arXiv2026

$\mathcal{O}_α$-transformation and its uncertainty principles

In this paper, we introduce a family of integral transforms, denoted by $\mathcal{O}_α$, and constructed via kernel fusion of the fractional Fourier transform (FRFT) with angle $α\notin π\mathbb{Z}$. We demonstrate that the $\mathcal{O}_α$-transformation constitutes a well-defined integral operator by establishing its basic operational properties. Besides, we survey various mathematical aspects of the uncertainty principles for the $\mathcal{O}_α$-transform, including Heisenberg's inequality, logarithmic uncertainty inequality, local uncertainty inequality, Hardy's inequality, Pitt's inequality, and Beurling-H{ö}rmander's theorem.

math.CA↗

arXiv2026

A refined variant of Hartley convolution: algebraic structures, spectral radius and related issues

In this work, we propose a novel convolution product associated with the $\mathscr{H}$-transform, denoted by $\underset{\mathscr{H}}{\ast}$, and explore its fundamental properties. Here, the $\mathscr{H}$-transform may be regarded as a refined variant of the classical Fourier, Hartley transform, with kernel function depending on two parameters $a,b$. Our first contribution shows that the space of integrable functions, equipped with multiplication given by the $\underset{\mathscr{H}}{\ast}$-convolution, constitutes the commutative Banach algebra over the complex field, albeit without an identity element. Second, establishes the Wiener--Lévy type invertibility criterion for $\mathscr{H}$-algebras, obtained through the density property and process of unitarization, which serves as a key step toward the proof of Gelfand's spectral radius theorem. Third, provides an explicit upper-bound of Young's inequality for $\underset{\mathscr{H}}{\ast}$-convolution and its direct corollary. Finally, all of these theoretical findings are applied to analyze specific classes of the Fredholm integral equations and heat source problems, yielding a priori estimates under the established assumptions.

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arXiv2025

Fundamental inequalities for the iterated Fourier-cosine convolution with Gaussian weight and its application

Derived from the results in [Giang et al.: \emph{Convolutions for the Fourier transforms with geometric variables and applications}, Math. Nachr. 283(12) (2010), 1758--1770], in this paper, we devoted to studying the boundedness properties for the Fourier-cosine convolution weighted by a Gaussian function of the form $γ=\exp(-\frac{1}{2}y^2)$ via Young's type theorem and Saitoh's type inequality. New norm estimations in the weighted space are obtained, and the application of the corresponding class of convolutions in Fredholm's second kind of integral equation is discussed. The conditions for the solvability of this equation on the $L_1$ space are also found, along with the analysis of an illustrative numerical example, which exemplifies that the present object and method solve cases that are not under the conditions of previously known techniques.

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arXiv2025

Upper bound coefficient for convolution structure associated to Hartley--Bessel transform

This paper is devoted to the study of a convolution structure denoted by $*_{\alpha}$, which is defined via the Hartley--Bessel transform. This concept was introduced in a recent work by F. Bouzeffour [\emph{J. Pseudo-Differ. Oper. Appl.}, 2024;15, Article 42]. We establish an analog of the Hausdorff--Young inequality for the Hartley--Bessel transform and convolution operator $*_{\alpha}$. This leads to the convolution $*_{\alpha}$ being uniformly bounded on the dual space. Moreover, in some special cases, our results yield a better upper bound coefficient for the convolution $*_{\alpha}$ than those previously obtained by Bouzeffour's result in [Theorem 4.4, \emph{J. Pseudo-Differ. Oper. Appl.}, 2024;15, Article 42]. Finally, we apply the convolution structure $*_{\alpha}$ to study the solvability of a particular class of integral equations and provide a priori estimates for solutions under appropriate conditions.

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arXiv2025

Composition structure of polyconvolution associated with index Kontorovich-Lebedev transform and Fourier integrals

Using Kakichev's classical concept and extending Yakubovich-Britvina's approach (\textit{Results. Math.} 55(1-2):175-197, 2009) and (\textit{Integral Transforms Spec. Funct.} 21(4):259--276, 2010) for setting up Kontorovich-Lebedev convolution operators, this paper proposes a new polyconvolution structure associated with the KL-transform and Fourier integrals. Our main contributions include demonstrating a one-dimensional Watson-type transform, providing necessary and sufficient conditions for this transform to serve as unitary on $L_2(\mathbb{R}_+)$, and inferring its inverse operator in symmetric form. The existence of this structure over specific function spaces and its connection with previously known convolutions are pointed out. Establish the Plancherel-type theorem, prove the convergence in the mean-square sense in $L_2(\mathbb{R}_+)$, and prove the boundedness of dual spaces via Riesz-Thorin's theorem. Derives new weighted $L_p$-norm inequalities and boundedness in a three-parametric family of Lebesgue spaces. These theoretical findings are applied to solve specific classes of the Toeplitz-Hankel equation, providing a priori estimations based on the established conditions for $L_1$ solvability.

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arXiv2024

Analog version of Hausdorff--Young's theorem for quadratic Fourier transforms and boundedness of oscillatory integral operator

The purpose of this paper is twofold. The first aim is based on Riesz--Thorin's interpolation theorem, we prove new Hausdorff--Young type inequalities for the Quadratic Fourier transforms in (Ann. Funct. Anal. 2014;5(1):10--23) and linear canonical transforms in (Mediterr. J. Math. 2018;15,13), which were introduced by Castro et al. The second aim is to investigate the boundedness of the oscillatory integral operator with polynomial phases, which is also presented in the last section of the article.

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arXiv2024

Trigonometric weighted generalized convolution operator associated with Fourier cosine-sine and Kontorovich-Lebedev transformations

The main objective of this work is to introduce the generalized convolution with trigonometric weighted $γ=\sin y$ involving the Fourier cosine-sine and Kontorovich-Lebedev transforms, and to study its fundamental results. We establish boundedness properties in a two-parametric family of Lebesgue spaces for this convolution operator. Norm estimation in the weighted $ L_p$ space is obtained and applications of the corresponding class of convolution integro-differential equations are discussed. The conditions for the solvability of these equations in $L_1$ space are also founded.

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arXiv2023

New polyconvolution product for Fourier-cosine and Laplace integral operators and their applications

The goal of this paper is to introduce the notion of polyconvolution for Fourier-cosine, Laplace integral operators, and its applications. The structure of this polyconvolution operator and associated integral transforms are investigated in detail. The Watson-type theorem is given, to establish necessary and sufficient conditions for this operator to be isometric isomorphism (unitary) on $L_2 (\mathbb{R}_+)$, and to get its inverse represented in the conjugate symmetric form. The correlation between the existence of polyconvolution with some weighted spaces is shown, and Young's type theorem, as well as the norm-inequalities in weighted space, are also obtained. As applications, we investigate the solvability of a class of Toeplitz plus Hankel type integral equations and linear Barbashin's equations with the help of factorization identities of such polyconvolution. Several examples are provided to illustrate the obtained results to ensure their validity and applicability.

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arXiv2023

Boundedness in $L_p$ spaces for the Hartley-Fourier convolutions operator and their applications

The paper deals with $L_p$-boundedness of the Hartley-Fourier convolutions operator and their applied aspects. We establish various new Young-type inequalities and obtain the structure of a normed ring in Banach space when equipping it with such convolutional multiplication. Weighted $L_p$-norm inequalities of these convolutions are also considered. As applications, we investigate the solvability and the bounded $L_1$-solution of a class of Fredholm-type integral equations and linear Barbashin's equations with the help of factorization identities of such convolutions. Several examples are provided to illustrate the obtained results to ensure their validity and applicability.

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