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Classical PLS-spaces: spaces of distributions, real analytic functions and their relatives

Paweł Domański (2004)

Banach Center Publications

This paper is an extended version of an invited talk presented during the Orlicz Centenary Conference (Poznań, 2003). It contains a brief survey of applications to classical problems of analysis of the theory of the so-called PLS-spaces (in particular, spaces of distributions and real analytic functions). Sequential representations of the spaces and the theory of the functor Proj¹ are applied to questions like solvability of linear partial differential equations, existence of a solution depending...

Continuation of holomorphic solutions to convolution equations in complex domains

Ryuichi Ishimura, Jun-ichi Okada, Yasunori Okada (2000)

Annales Polonici Mathematici

For an analytic functional S on ℂ n , we study the homogeneous convolution equation S * f = 0 with the holomorphic function f defined on an open set in ℂ n . We determine the directions in which every solution can be continued analytically, by using the characteristic set.

Exponential convergence of hp quadrature for integral operators with Gevrey kernels

Alexey Chernov, Tobias von Petersdorff, Christoph Schwab (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

Galerkin discretizations of integral equations in ℝ d require the evaluation of integrals I = ∫ S ( 1 ) ∫ S ( 2 ) g ( x , y ) d y d x where S(1),S(2) are d-simplices and g has a singularity at x = y. We assume that g is Gevrey smooth for x ≠ y and satisfies bounds for the derivatives which allow algebraic singularities at x = y. This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules 𝒬 N using N function evaluations of g which achieves exponential convergence |I – 𝒬 N | ≤C exp(–rNγ) with...

Exponential convergence of hp quadrature for integral operators with Gevrey kernels

Alexey Chernov, Tobias von Petersdorff, Christoph Schwab (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

Galerkin discretizations of integral equations in ℝ d require the evaluation of integrals I = ∫ S ( 1 ) ∫ S ( 2 ) g ( x , y ) d y d x where S(1),S(2) are d-simplices and g has a singularity at x = y. We assume that g is Gevrey smooth for x ≠ y and satisfies bounds for the derivatives which allow algebraic singularities at x = y. This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules 𝒬 N using N function evaluations of g which achieves exponential convergence |I – 𝒬 N | ≤C exp(–rNγ) with...

First kind integral equations for the numerical solution of the plane Dirichlet problem

Søren Christiansen (1989)

Aplikace matematiky

We present, in a uniform manner, several integral equations of the first kind for the solution of the two-dimensional interior Dirichlet boundary value problem. We apply a general numerical collocation method to the various equations, and thereby we compare the various integral equations, and recommend two of them. We give a survey of the various numerical methods, and present a simple method for the numerical solution of the recommended integral equations.

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