Approximation Methods for the Riemann-Hilbert Problem
Frontiers in mathematics · 2008
Approximation methods for the Riemann-Hilbert boundary problem are considered in this chapter. A particular feature of these problems is that the operators studied act in a pair of spaces, so the corresponding operator spaces do not have any multiplication operation which makes the use of algebraic techniques more difficult. However, by introducing appropriate para-algebras, one can obtain necessary and sufficient stability conditions for the approximation methods considered. Note that there are two formulations for the Riemann-Hilbert boundary problem: one is concerned with the solutions from a chosen space of functions analytic inside a given domain D of the complex plane ℂ, whereas another contains an additional requirement that the corresponding solutions take real values at a fixed point of the domain D . Here, we deal with the second formulation because it arises more often in applications (cf. [162, Problem P]).