Explicit calculation of singular integrals of tensorial polyadic kernels

Abstract

The Riesz transform of u u : S ( R n ) → S ′ ( R n ) \mathcal {S}(\mathbb {R}^n) \rightarrow \mathcal {S’}(\mathbb {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 S ( R n ) → S ′ ( R n × n × … n ) \mathcal {S}(\mathbb {R}^n) \rightarrow \mathcal {S’}(\mathbb {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 n=2 , with application to image analysis.