An approximation for a subclass of the Riemann-Hilbert problems

摘要

Consider the problem of solving a Riemann-Hilbert problem with 'zero index'. Abraham (2000, IMA J. Appl. Math., 65, 257-281) suggested to replace a possibly complicated kernel of a homogeneous Riemann-Hilbert problem with a Pade approximant that uniformly approximates the original kernel. Abraham's procedure fails whenever the kernel cannot be approximated uniformly by a Pade approximant (see Example 1). This article (i) provides an approximation technique to approximate solutions of a non-homogeneous Riemann-Hilbert problem with zero index in L-p(R) (1 < p < infinity) sense, which improves the result by Abraham in two directions (weaker conditions on approximating functions and solutions for a non-homogeneous Riemann-Hilbert problem with zero index). Also, we discussed an interesting case p = infinity (uniformly approximation). (ii) Using the Egoroff's theorem provides a pointwise approximate solutions for a class of non-homogeneous Riemann-Hilbert problem with zero index. (iii) Using the Shannon sampling theorem provides explicit solutions for certain non-homogeneous Riemann-Hilbert problems with zero index. Some approximations which exploiting this fact will be discussed. (iv) Applications to integral equations are given.