A Faber-Krahn inequality for wavelet transforms

TL;DR

This paper establishes a Faber-Krahn type inequality for wavelet transforms, showing that optimal concentration occurs on hyperbolic discs, using hyperbolic geometry and rearrangement techniques.

Contribution

It proves that wavelet transform concentration is maximized on hyperbolic discs for special windows, answering a question by Abreu and Dörfler.

Findings

Optimal concentration sets are hyperbolic discs.

Uses hyperbolic rearrangement and isoperimetric inequalities.

Answers a previously open question.

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.