Numerical conformal mapping via the Bergman kernel using the generalized minimum residual method

AbstractThe Bergman kernel function is known to satisfy a certain boundary integral equation of the second kind. For boundaries that possess symmetrical qualities, the integral equation can be transformed into another integral equation that uses only a small part of the original boundary. This paper applies an iterative procedure known as the generalized minimum residual method for the computation of the Riemann mapping function via the Bergman kernel. The complexity of this procedure is O(n2), where n is the number of collocation points on the boundary of the region. Numerical implementation on some test regions is also presented

Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems

We consider solution operators of linear ordinary boundary problems with "too many" boundary conditions, which are not always solvable. These generalized Green's operators are a certain kind of generalized inverses of differential operators. We answer the question when the product of two generalized Green's operators is again a generalized Green's operator for the product of the corresponding differential operators and which boundary problem it solves. Moreover, we show that---provided a factorization of the underlying differential operator---a generalized boundary problem can be factored into lower order problems corresponding to a factorization of the respective Green's operators. We illustrate our results by examples using the Maple package IntDiffOp, where the presented algorithms are implemented.Comment: 19 page

CLT in Functional Linear Regression Models

International audienceWe propose in this work to derive a CLT in the functional linear regression model to get confidence sets for prediction based on functional linear regression. The main difficulty is due to the fact that estimation of the functional parameter leads to a kind of ill-posed inverse problem. We consider estimators that belong to a large class of regularizing methods and we first show that, contrary to the multivariate case, it is not possible to state a CLT in the topology of the considered functional space. However, we show that we can get a CLT for the weak topology under mild hypotheses and in particular without assuming any strong assumptions on the decay of the eigenvalues of the covariance operator. Rates of convergence depend on the smoothness of the functional coefficient and on the point in which the prediction is made

Some exponentially decreasing error bounds for a numerical inversion of the Laplace transform

AbstractConvergence properties of a class of least-squares methods for finding approximate inverses of the Laplace transform are obtained by using reproducing kernel Hilbert space techniques (or, alternatively, related minimization techniques) and some classical interpolation results

Average sampling in L2

10.1016/j.crma.2009.07.011Comptes Rendus Mathematique34717-181007-101

Biorthogonality and reproducing property

Integral operators involving the Szego, the Bergman and the Cauchy kernels are known to have the reproducing property, i.e. all of them reproduce holomorphic functions. Both the Szego and the Bergman kernels have series representations in terms of orthonormal basis. In this paper we derive the Cauchy kernel by means of biorthogonality. The ideas involved are then applied to construct a non-Hermitian kernel admitting reproducing property for a space associated with the Bergman kernel

Numerical conformal mapping for symmetric regions via the Bergman kernel

The Bergman kernel functions is known to satisfy a certain boundary integral equation of the second kind. For boundaries that possess some symmetry, it is shown how to transform the integral equation into another integral equation that uses only a small part of the original boundary. This provides an efficient computation of the Riemann mapping function for symmetric regions via the Bergman kernel