Abstract
The Wiener–Hopf factorization of 2 × 2 matrix functions and its close relation to scalar Riemann–Hilbert problems on Riemann surfaces is investigated. A family of function classes denoted C ( Q 1 , Q 2 ) is defined. To each class C ( Q 1 , Q 2 ) a Riemann surface Σ is associated, so that the factorization of the elements of C ( Q 1 , Q 2 ) is reduced to solving a scalar Riemann–Hilbert problem on Σ. For the solution of this problem, a notion of Σ-factorization is introduced and a factorization theorem is presented. An example of the factorization of a function belonging to the group of exponentials of rational functions is studied. This example may be seen as typical of applications of the results of this paper to finite-dimensional integrable systems.
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