PERRIN, Mathias; GRUY, Frédéric

doi:10.1090/qam/1629

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PERRIN, Mathias
Laboratoire Ondes et Matière d'Aquitaine [LOMA]
Université de Bordeaux [UB]

GRUY, Frédéric
Laboratoire Georges Friedel [LGF-ENSMSE]
Centre Sciences des Processus Industriels et Naturels [SPIN-ENSMSE]

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Quarterly of Applied Mathematics. 2022, vol. 81, n° 1, p. 65 - 86

American Mathematical Society

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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.< Réduire

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Explicit calculation of singular integrals of tensorial polyadic kernels

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