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IMA Journal of Applied Mathematics Advance Access originally published online on November 6, 2008
IMA Journal of Applied Mathematics 2009 74(4):533-547; doi:10.1093/imamat/hxn034
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

An approximation for a subclass of the Riemann–Hilbert problems

Dan Kucerovsky

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3

Amir T. Payandeh Najafabadi{dagger}

Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, Tehran 1983963113, Iran

{dagger} Email: amirtpayandeh{at}sbu.ac.ir

Received on May 8, 2008; Revision received July 30, 2008. Accepted on September 8, 2008

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 Padé approximant that uniformly approximates the original kernel. Abraham's procedure fails whenever the kernel cannot be approximated uniformly by a Padé 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 Lp(R) (1 < p < {infty}) 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 = {infty} (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.

Keywords: boundary-value problem; sectionally analytic function; principal-value integral; Hölder condition; winding number; radial limit; Fourier transform; Hilbert transform; Padé approximant; continued fraction expansion; Titchmarsh–Riesz theorem; convolution theorem; Hausdorff–Young theorem; Shannon sampling theorem; Wiener–Hopf integral equation.


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