Abstract
In this paper, the Wiener–Hopf factorization problem is presented in a unified framework with the Riemann–Hilbert factorization. This allows to establish the exact relationship between the two types of factorization. In particular, in the Wiener–Hopf problem one assumes more regularity than for the Riemann–Hilbert problem. It is shown that Wiener–Hopf factorization can be obtained using Riemann– Hilbert factorization on certain lines.
Highlights
The Wiener–Hopf and the Riemann–Hilbert problems are a subject of many books and articles (Câmara & dos Santos, 1999; Ehrhardt & Speck, 2002; Èrkhardt & Spitkovskiı, 2001; Gohberg et al, 2003; Rogosin & Mishuris)
To the author’s knowledge there was no systematic study of the exact relationship of the two methods. To fill this gap is the purpose of this article. It has been suggested in Noble (1958, Chapter 4.2) that the Wiener–Hopf equation are a special case of a Riemann–Hilbert equation
The Riemann–Hilbert problem connects boundary values of two analytic functions on a contour and the Wiener–Hopf equation is defined on the strip of common analyticity of two functions
Summary
The Wiener–Hopf and the Riemann–Hilbert problems are a subject of many books and articles (Câmara & dos Santos, 1999; Ehrhardt & Speck, 2002; Èrkhardt & Spitkovskiı, 2001; Gohberg et al, 2003; Rogosin & Mishuris). In a different book (Gakhov & Cerskiı, 1978, Chapter 14.4), it has been stated that the Wiener– Hopf equations results from a bad choice of functions spaces and instead a Riemann–Hilbert equations should be considered. RELATIONSHIP BETWEEN A WIENER–HOPF AND A RIEMANN–HILBERT PROBLEM find two factors F+(t) and F−(t), which have analytic extensions in the upper and lower half-planes respectively In this rare case, the factorization can be obtained by inspection. This allows to prove theorems about the exact relationship of Wiener–Hopf and Riemann–Hilbert factorization
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