Composition structure of polyconvolution associated with index Kontorovich-Lebedev transform and Fourier integrals
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866915402275094528 |
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| author | Tuan, Trinh |
| author_facet | Tuan, Trinh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10243 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Composition structure of polyconvolution associated with index Kontorovich-Lebedev transform and Fourier integrals Tuan, Trinh Classical Analysis and ODEs Functional Analysis 42A38, 44A20, 44A35, 45A05 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. |
| title | Composition structure of polyconvolution associated with index Kontorovich-Lebedev transform and Fourier integrals |
| topic | Classical Analysis and ODEs Functional Analysis 42A38, 44A20, 44A35, 45A05 |
| url | https://arxiv.org/abs/2503.10243 |