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October 13, 2025Open Access

Hardy's Theorem for the (k, 2n) -Fourier Transform

Key Points

A Hardy-type uncertainty principle is established through direct comparison with Gaussian functions, highlighting its significance.
A Cowling-Price-type theorem is extended to the (k, 2/n)-Fourier transform, proving its applicability in broader contexts.
Optimal cases are identified, providing deeper insights into the nature of these theorems and their implications.
A dynamical version of Hardy's theorem demonstrates how the uncertainty principle evolves over time within the heat equation framework.

Abstract

By comparing a function and its (k, 2n) -Fourier transform to a Gaussian analogue, we establish a Hardy-type uncertainty principle. We extend these results to an Lᵖ-Lq framework, proving a Cowling-Price-type theorem for the (k, 2n) -Fourier transform. Optimal cases are identified and discussed in detail for both theorems. Furthermore, we investigate the heat equation in this context, deriving a dynamical version of Hardy's theorem that illustrates the temporal evolution of the uncertainty principle in this generalized setting.

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Cite This Study

Jilani et al. (Sun,) studied this question.

synapsesocial.com/papers/68ece2abd1bb2827d12971b4 https://doi.org/https://doi.org/10.48550/arxiv.2503.01094

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