The Wiener--Hopf equation and Blaschke products

V. B. Dybin

Sbornik. Mathematics, Tome 70 (1991) no. 1, pp. 205-230

Résumé

A Wiener–Hopf operator $A$ is studied in the space of functions locally square-integrable on $\mathbf R$ and slowly increasing to $\infty$. The symbol of the operator is an infinitely differentiable function on $\mathbf R$ and has at $\infty$ a discontinuity of “vorticity point” type described either by a Blaschke function with all its zeros concentrated in a strip and bounded away from $\mathbf R$, or by an outer function meromorphic in the complex plane with separated set of real zeros of bounded multiplicity. The operator $A$ is one-sidedly invertible, and $\operatorname{ind}A=\pm\infty$. Procedures are worked out for inverting it. The subspace $\operatorname{ker}A$ is described in terms of generalized Dirichlet series.