A Faber–Krahn inequality for Wavelet transforms

João P. G. Ramos, Paolo Tilli

2023Bulletin of the London Mathematical Society13 citationsDOI Open Access PDF

Abstract

Abstract For some special window functions , we prove that, over all sets of fixed hyperbolic measure , those for which the Wavelet transform with window concentrates optimally are exactly the discs with respect to the pseudo‐hyperbolic metric of the upper half space. This answers a question raised by Abreu and Dörfler in Abreu and Dörfler ( Inverse Problems 28 (2012) 16). Our techniques make use of a framework recently developed by Nicola and Tilli in Nicola and Tilli ( Invent. Math . 230 (2022) 1–30), 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.

Topics & Concepts

MathematicsIsoperimetric inequalityHyperbolic spaceMetric (unit)Dilation (metric space)Context (archaeology)Pure mathematicsWaveletInverseMathematical analysisCombinatoricsGeometryPaleontologyBiologyArtificial intelligenceOperations managementEconomicsComputer scienceMathematical Analysis and Transform MethodsNumerical methods in inverse problemsAnalytic and geometric function theory