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Mittag-Leffler functions in the Fourier space by Abdelhakim, Ahmed A. is a scholarly article available to read on EtoBox.

What is Mittag-Leffler functions in the Fourier space about?

Let $\alpha \in (0,2)$ and let $\beta>0$. Fix $-\pi<\varphi\leq \pi$ such that $|\varphi|>\alpha \pi/2$. We determine the precise asymptotic behaviour of the Fourier transform of $E_{\alpha,\beta}(e^{\dot{\imath} \varphi} |\cdot|^{\sigma})$ whenever $\sigma>(n-1)/2$. Remarkably, this asymptotic behaviour turns out to be independent of $\alpha$ and $\beta$. This helps us determine the values of the Lebesgue exponent $p=p(\sigma)$, $\sigma>(n-1)/2$, for which $\mathcal{F} \left(E_{\alpha,\beta}(e^{\dot{\imath} \varphi} |\cdot|^{\sigma})\right)$ is in $L^{p}(\mathbb{R}^{n})$. These values cannot be obtained via the Hausdorff-Young inequality. This problem arises in the study of space-time fractional equations. Our approach provides an effective alternative to the asymptotic analysis of the Fox $H-$ functions recently applied to the case $\alpha \in (0,1)$, $\beta=\alpha, 1$, $\varphi=-\pi/2,\pi$. We rather rely on an appropriate integral representative that represents $E_{\alpha,\beta}$ continuously up to the origin, and we develop an extended asymptotic expansion for the Bessel function whose coefficients and remainder term are obtained explicitly.

Author

Abdelhakim, Ahmed A.

Published

2024

Language

EN