On the asymptotic behaviour of the Fourier transform of the Mittag-Leffler function

2024/02/07 by Ahmed A. Abdelhakim, Abdelhakim, Ahmed A.
Computer Science · Mathematics · #30E20 #33E12 #34E05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods in inverse problems #Primary: 30E15 #Secondary: 42B10 #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2402.05230

openalex publication_date 2024/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let α∈ (0,2) and let β>0. Fix -π<φ≤ π such that |φ|>απ/2. We obtain asymptotic upper bounds on the Fourier transform of the radially symmetric tempered distribution ℝⁿ\ni x↦ E_α,β(e^\imath φ |x|^σ), 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(ℝⁿ). Such values cannot be obtained via the well-known L^p(ℝⁿ) properties of E_α,β and the Hausdorff-Young inequality, when σ≤ n/2.

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