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arXiv · 2205.07998

A Faber-Krahn inequality for wavelet transforms

João P. G. Ramos·Paolo Tilli
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Abstract

For some special window functions $\psi_{\beta} \in H^2(\mathbb{C}^+),$ we prove that, over all sets $\Delta \subset \mathbb{C}^+$ of fixed hyperbolic measure $\nu(\Delta),$ the ones over which the Wavelet transform $W_{\overline{\psi_{\beta}}}$ with window $\overline{\psi_{\beta}}$ concentrates optimally are exactly the discs with respect to the pseudohyperbolic metric of the upper half space. This answers a question raised by Abreu and D\"orfler. Our techniques make use of a framework recently developed in a previous work by F. Nicola and the second author, but in the hyperbolic context induced by the dilation symmetry of the Wavelet transform. This leads us naturally to use a hyperbolic rearrangement function, as well as the hyperbolic isoperimetric inequality, in our analysis.

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João P. G. Ramos, Paolo Tilli. 2022-05-16. A Faber-Krahn inequality for wavelet transforms. https://arxiv.org/abs/2205.07998

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