An approximation for a subclass of the Riemann–Hilbert problems

IMA Journal of Applied Mathematics

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 Pad´e approximant that uniformly approximates the original kernel.
Abraham’s procedure fails whenever the kernel cannot be approximated uniformly by a Pad´e 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 < ∞) 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 = ∞ (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.