On the asymptotic behaviour of the Fourier transform of the Mittag-Leffler function
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911623911833600 |
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| author | Abdelhakim, Ahmed A. |
| author_facet | Abdelhakim, Ahmed A. |
| contents | Let $α\in (0,2)$ and let $β>0$. Fix $-π<φ\leq π$ such that $|φ|>απ/2$. We obtain asymptotic upper bounds on the Fourier transform of the radially symmetric tempered distribution \begin{equation*} \mathbb{R}^n\ni x\mapsto E_{α,β}(e^{\dot{\imath} φ} |x|^σ), \end{equation*} for $σ>(n-1)/2$, where $E_{α,β}$ is the two-parameter Mittag-Leffler function. As an application, we obtain some values of the Lebesgue exponent $p=p(σ)$, $σ>(n-1)/2$, for which the Fourier transform is in $L^{p}(\mathbb{R}^{n})$. Such values cannot be obtained via the well-known $L^{p}(\mathbb{R}^{n})$ properties of $E_{α,β}$ and the Hausdorff-Young inequality, when $σ\leq n/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05230 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the asymptotic behaviour of the Fourier transform of the Mittag-Leffler function Abdelhakim, Ahmed A. Classical Analysis and ODEs Analysis of PDEs Primary: 30E15, 33E12, Secondary: 42B10, 30E20, 34E05 Let $α\in (0,2)$ and let $β>0$. Fix $-π<φ\leq π$ such that $|φ|>απ/2$. We obtain asymptotic upper bounds on the Fourier transform of the radially symmetric tempered distribution \begin{equation*} \mathbb{R}^n\ni x\mapsto E_{α,β}(e^{\dot{\imath} φ} |x|^σ), \end{equation*} for $σ>(n-1)/2$, where $E_{α,β}$ is the two-parameter Mittag-Leffler function. As an application, we obtain some values of the Lebesgue exponent $p=p(σ)$, $σ>(n-1)/2$, for which the Fourier transform is in $L^{p}(\mathbb{R}^{n})$. Such values cannot be obtained via the well-known $L^{p}(\mathbb{R}^{n})$ properties of $E_{α,β}$ and the Hausdorff-Young inequality, when $σ\leq n/2$. |
| title | On the asymptotic behaviour of the Fourier transform of the Mittag-Leffler function |
| topic | Classical Analysis and ODEs Analysis of PDEs Primary: 30E15, 33E12, Secondary: 42B10, 30E20, 34E05 |
| url | https://arxiv.org/abs/2402.05230 |