The Wiener-Hopf Equation in the Nevanlinna and Smirnov Algebras and Ultra-Distributions

Abstract

1. The Wiener-Hopf equation on the semi-axis

$$\phi \left( \xi \right) = \int\limits_0^\infty {{\text{k}}\left( {\xi - \xi \prime} \right)\phi \left( {\xi \prime} \right){\text{d}}\xi \prime + {\text{f}}\left( \xi \right),\quad \xi \mathop > \limits_ = 0} $$

(1.1)

and the associated Riemann-Hilbert problem on a real axis

$$\rho \left( {\text{x}} \right)\phi ^ + \left( {\text{x}} \right) = \psi ^ - \left( {\text{x}} \right) + {\text{F}}\left( {\text{x}} \right)\quad {\text{a}}{\text{.e}}{\text{.}}\;{\text{on}}\;\mathbb{R}$$

(1.2)

has been investigated by many mathematicians starting from N. Wiener and E. Hopf [1] under various assumptions about kernel k and function ρ. An important contribution to their theory has been made by V. A. Fok [2], N. I. Muschelishvili [3, 4], I. N. Vekua [24], N. P. Vekua [3, 5], V. A. Ambartsumian [6], F. D. Gahov [7], S. Chandrasekhar [8], V. V. Sobolev [9], M. G. Krein [10, 11], I. I. Daniluk [26], B. V. Bojarskii [27], I. B. Simonenko [28], G. S. Litvinchuk [29], M. V. Maslennikov [12], N. B. Engibarjan [13], V. M. Kokilashvili and V. A. Paatashvili [30] and others.