Multidimensional quadratic-phase Fourier transform and its uncertainty principles

Abstract

The main aim of this article is to propose a multidimensional quadratic-phase Fourier transform (MQFT) that generalises the well-known and recently introduced quadratic-phase Fourier transform (as well as, of course, the Fourier transform itself) to higher dimensions. In addition to the definition itself, some crucial properties of this new integral transform will be deduced. These include a Riemann-Lebesgue lemma for the MQFT, a Plancherel lemma for the MQFT and a Hausdorff-Young inequality for the MQFT. A second central objective consists of obtaining different uncertainty principles for this MQFT. To this end, using techniques that include obtaining various auxiliary inequalities, the study culminates in the deduction of $L^p$-type Heisenberg-Pauli-Weyl uncertainty principles and $L^p$-type Donoho-Stark uncertainty principles for the MQFT.

Keywords

• multidimensional quadratic-phase Fourier transform
• Donoho-Stark uncertainty principle
• Heisenberg-Pauli-Weyl uncertainty principle
• Riemann-Lebesgue lemma
• Hausdorff-Young inequality

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