Some compact and non-compact embedding theorems for the function spaces defined by fractional Fourier transform

Year 2021, Volume: 50 Issue: 6, 1620 - 1635, 14.12.2021

Erdem Toksoy , Ayşe Sandıkçı

https://doi.org/10.15672/hujms.795924

Abstract

The fractional Fourier transform is a generalization of the classical Fourier transform through an angular parameter $\alpha $. This transform uses in quantum optics and quantum wave field reconstruction, also its application provides solving some differrential equations which arise in quantum mechanics. The aim of this work is to discuss compact and non-compact embeddings between the spaces $A_{\alpha ,p}^{w,\omega }\left(\mathbb{R}^{d}\right) $ which are the set of functions in ${L_{w}^{1}\left(\mathbb{R}^{d}\right) }$ whose fractional Fourier transform are in ${L_{\omega}^{p}\left(\mathbb{R}^{d}\right) }$. Moreover, some relevant counterexamples are indicated.

Keywords

Fractional Fourirer transform, weighted Lebesgue spaces, compact embedding

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