Equations with Dual Convolution Wiener–Hopf Operators

L. G. Salekhov^a, L. L. Salekhova<sup>b

a</sup> Kazan State University
^b Kazan State Pedagogical University

Abstract: A class of equations with dual convolution Wiener–Hopf operators is considered. The investigation is carried out in the space of generalized functions, admitting analytical presentations (Cauchy presentations). Equations of the considered class are equivalent to a boundary value problem regarding the disappearing at infinity piecewise-holomorphic function $\widehat\Phi(z)=(\widehat\Phi^+(z),\widehat\Phi^-(z))$. Boundary condition is given on the real axis $\mathbb R$ and is understood in the sense of generalized functions. The considered equations are reduced to some isomorphic equations by means of Fourier transformation in space of tempered generalized functions. The latter, according to hypothesis of generalized functions regularity, include bilateral and unilateral Wiener–Hopf equations, dual integral equations with constant and variable limits, and dual ordinary differential equations.

Keywords: dual convolution operator, generalized function, analytical presentation, partly-holomorphic function, dual integral equations, dual ordinary differential equations.

UDC: 517.982

Received: 22.01.2008

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