Integrability properties of quasi-regular representations of N A groups

Jordy Timo van Velthoven

doi:10.5802/crmath.372

Integrability properties of quasi-regular representations of N A groups

Jordy Timo van Velthoven^*

^*Corresponding author for this work

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Abstract

Let G = N ⋉ A, where N is a graded Lie group and A = R⁺ acts on N via homogeneous dilations. The quasi-regular representation π = ind^G_A(1) of G can be realised to act on L²(N). It is shown that for a class of analysing vectors the associated wavelet transform defines an isometry from L²(N) into L²(G) and that the integral kernel of the corresponding orthogonal projector has polynomial off-diagonal decay. The obtained reproducing formula is instrumental for obtaining decomposition theorems for function spaces on nilpotent groups.

Original language	English
Pages (from-to)	1125-1134
Number of pages	10
Journal	Comptes Rendus Mathematique
Volume	360
DOIs	https://doi.org/10.5802/crmath.372
Publication status	Published - 2022

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CRMATH_2022__360_G10_1125_0Final published version, 586 KBLicence: CC BY

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abstract = "Let G = N ⋉ A, where N is a graded Lie group and A = R+ acts on N via homogeneous dilations. The quasi-regular representation π = indGA(1) of G can be realised to act on L2(N). It is shown that for a class of analysing vectors the associated wavelet transform defines an isometry from L2(N) into L2(G) and that the integral kernel of the corresponding orthogonal projector has polynomial off-diagonal decay. The obtained reproducing formula is instrumental for obtaining decomposition theorems for function spaces on nilpotent groups.",

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AB - Let G = N ⋉ A, where N is a graded Lie group and A = R+ acts on N via homogeneous dilations. The quasi-regular representation π = indGA(1) of G can be realised to act on L2(N). It is shown that for a class of analysing vectors the associated wavelet transform defines an isometry from L2(N) into L2(G) and that the integral kernel of the corresponding orthogonal projector has polynomial off-diagonal decay. The obtained reproducing formula is instrumental for obtaining decomposition theorems for function spaces on nilpotent groups.

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Integrability properties of quasi-regular representations of N A groups

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