Convergence of singular integrals with general measures

We show that L2-bounded singular integrals in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost everywhere existence of principal values.

On the boundedness on inhomogeneous Lipschitz spaces of fractional integrals, singular integrals and hypersingular integrals associated to non-doubling measures on metric spaces

In this paper we prove T1 type necessary and sufficient conditions for the boundedness on inhomogeneous Lipschitz spaces of fractional integrals and singular integrals defined on a measure metric space whose measure satisfies a n-dimensional growth. We also show that hypersingular integrals are bounded on these spaces.

Singular integrals on Ahlfors-David regular subsets of the Heisenberg group

"Vegeu el resum a l'inici del document del fitxer adjunt."

A characterization of two weight norm inequalities for maximal singular integrals

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An efficient nonlinear transformation for the numerical computation of the singular integrals appearing in the 2-D Boundary Element Method

We discuss several methods, based on coordinate transformations, for the evaluation of singular and quasisingular integrals in the direct Boundary Element Method. An intrinsec error of some of these methods is detected. Two new transformations are suggested which improve on those currently available.

Sigmoidal cosine series on the interval

We construct a set of functions, say, psi([r])(n) composed of a cosine function and a sigmoidal transformation gamma(r) of order r > 0. The present functions are orthonormal with respect to a proper weight function on the interval [-1, 1]. It is proven that if a function f is continuous and piecewise smooth on [-1, 1] then its series expansion based on psi([r])(n) converges uniformly to f so long as the order of the sigmoidal transformation employed is 0 < r

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.

Remarks on methods for the computation of boundary-element integrals by co-ordinate transformation

It is well known that the evaluation of the influence matrices in the boundary-element method requires the computation of singular integrals. Quadrature formulae exist which are especially tailored to the specific nature of the singularity, i.e. log(*- x0)9 Ijx- JC0), etc. Clearly the nodes and weights of these formulae vary with the location Xo of the singular point. A drawback of this approach is that a given problem usually includes different types of singularities, and therefore a general-purpose code would have to include many alternative formulae to cater for all possible cases. Recently, several authors1"3 have suggested a type independent alternative technique based on the combination of standard Gaussian rules with non-linear co-ordinate transformations. The transformation approach is particularly appealing in connection with the p.adaptive version, where the location of the collocation points varies at each step of the refinement process. The purpose of this paper is to analyse the technique in eference 3. We show that this technique is asymptotically correct as the number of Gauss points increases. However, the method possesses a 'hidden' source of error that is analysed and can easily be removed.

A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary methods

The numerical strategies employed in the evaluation of singular integrals existing in the Cauchy principal value (CPV) sense are, undoubtedly, one of the key aspects which remarkably affect the performance and accuracy of the boundary element method (BEM). Thus, a new procedure, based upon a bi-cubic co-ordinate transformation and oriented towards the numerical evaluation of both the CPV integrals and some others which contain different types of singularity is developed. Both the ideas and some details involved in the proposed formulae are presented, obtaining rather simple and-attractive expressions for the numerical quadrature which are also easily embodied into existing BEM codes. Some illustrative examples which assess the stability and accuracy of the new formulae are included.

Rough Maximal Oscillatory Singular Integral Operators

2000 Mathematics Subject Classification: Primary 42B20; Secondary 42B15, 42B25

Analysis of stiffened plates by the boundary element method

In this paper, a formulation for representation of stiffeners in plane stress by the boundary elements method (BEM) in linear analysis is presented. The strategy is to adopt approximations for the displacements in the central line of the stiffener. With this simplification the Spurious oscillations in the stress along stiffeners with small thickness is prevented. Worked examples are analyzed to show the efficiency of these techniques, especially in the insertion of very narrow sub-regions, in which quasi-singular integrals are calculated, with stiffeners that are much stiffer than the main domain. The results obtained with this formulation are very close to those obtained with other formulations. (C) 2007 Elsevier Ltd. All rights reserved.

Error bounds for low-rank approximations of the first exponential integral kernel

A hierarchical matrix is an efficient data-sparse representation of a matrix, especially useful for large dimensional problems. It consists of low-rank subblocks leading to low memory requirements as well as inexpensive computational costs. In this work, we discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and, hence, it is weakly singular. We develop analytical expressions for the approximate degenerate kernels and deduce error upper bounds for these approximations. Some computational results illustrating the efficiency and robustness of the approach are presented.

Best approach regions for potential spaces

We characterize the approach regions so that the non-tangential maximal function is of weak-type on potential spaces, for which we use a simple argument involving Carleson measure estimates.