Singular integral

Résumé

In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : T(f)(x) = \int K(x,y)f(y) , dy, whose kernel function K : Rn×Rn → R is singular along the diagonal x = y. Specifically, the singularity is such that |K(x, y)| is of size |x − y|−n asymptotically as |x − y| → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over |y − x| > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp(Rn). The Hilbert transform Hilbert transform The archetypal singular integral operator is the Hilbert transform H. It is given by convolution

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https://en.wikipedia.org/wiki/Singular_integral

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Numerical analysis of a non-singular boundary integral method: Part I. The circular case

Jacques Rappaz

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we present here a method which consists in inverting, on a finite element space, a non-singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely define the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme. Furthermore, this numerical scheme does not give rise to any singular integral. The scheme can also be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single layer potential, the special quadrature formula being defined by the introduction of a neighbouring curve. In the present paper, we prove stability and we give error estimates of our numerical scheme when the Laplace problem is set on a disk. We will extend our results to any domains by using compact perturbation arguments, in a second paper. Copyright 2001 John Wiley & Sons, Ltd.

2001

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Numerical analysis of a non-singular boundary integral method: Part II: The general case

Jacques Rappaz

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we have presented in a previous paper a method which consists in inverting, on a finite element space, a non-singular integral operator for circular domains. This operator was described as a geometrical perturbation of the Steklov operator, and we have precisely defined the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme in which there are non-singular integrals. We have also presented another point of view under which the method can be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single-layer potential. In the present paper, we extend the results given in the previous paper to more general cases for which the Laplace problem is set on any (infinity) domains. We prove that the properties of stability and convergence remain valid. Copyright (C) 2002 John Wiley Sons, Ltd.

2002

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Harmonic singular integrals and steerable wavelets in L-2(R-d)

Michaël Unser, John Paul Ward

Here we present a method of constructing steerable wavelet frames in L-2(R-d) that generalizes and unifies previous approaches, including Simoncelli's pyramid and Riesz wavelets. The motivation for steerable wavelets is the need to more accurately account for the orientation of data. Such wavelets can be constructed by decomposing an isotropic mother wavelet into a finite collection of oriented mother wavelets. The key to this construction is that the angular decomposition is an isometry, whereby the new collection of wavelets maintains the frame bounds of the original one. The general method that we propose here is based on partitions of unity involving spherical harmonics. A fundamental aspect of this construction is that Fourier multipliers composed of spherical harmonics correspond to singular integrals in the spatial domain. Such transforms have been studied extensively in the field of harmonic analysis, and we take advantage of this wealth of knowledge to make the proposed construction practically feasible and computationally efficient. (C) 2013 Elsevier Inc. All rights reserved.

Academic Press Inc Elsevier Science2014

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