Article Dans Une Revue Quarterly of Applied Mathematics Année : 2022

Explicit calculation of singular integrals of tensorial polyadic kernels

Résumé

The Riesz transform of u $u$ : $\mathcal{S}(\R^n) \rightarrow \mathcal{S'}(\R^n)$ is defined as a convolution by a singular kernel, and can be conveniently expressed using the Fourier Transform and a simple multiplier. We extend this analysis to higher order Riesz transforms, i.e. some type of singular integrals that contain tensorial polyadic kernels and define an integral transform for functions $\mathcal{S}(\R^n) \rightarrow \mathcal{S'}(\R^{ n \times n \times \dots n})$. We show that the transformed kernel is also a polyadic tensor, and propose a general method to compute explicitely the Fourier mutliplier. Analytical results are given, as well as a recursive algorithm, to compute the coefficients of the transformed kernel. We compare the result to direct numerical evaluation, and discuss the case n = 2, with application to image analysis.

Fichier principal
Vignette du fichier
Fourier_Transform_Singular_Polyadic_Kernels.pdf (655.49 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Licence

Dates et versions

hal-03768198 , version 1 (02-09-2022)

Licence

Identifiants

Citer

Mathias Perrin, Frédéric Gruy. Explicit calculation of singular integrals of tensorial polyadic kernels. Quarterly of Applied Mathematics, 2022, 81 (1), pp.65 - 86. ⟨10.1090/qam/1629⟩. ⟨hal-03768198⟩
123 Consultations
156 Téléchargements

Altmetric

Partager

  • More