The two straight line approach for periodic diffraction boundary-value problems

Boundary-transmission problems of first order for the Helmholtz equation are considered within the context of wave diffraction by a periodic strip grating and formulated as convolution type operators acting on a Bessel potential periodic space setting. Two boundary-value problems are studied for an arbitrary geometry of the grating: the oblique derivative and the classic Neumann boundary-value problems. The convolution type operators on the grating which correspond to the given boundary-transmission problems are associated with Toeplitz operators acting on spaces of matrix functions defined on composed contours. A Fredholm theory for periodic boundary-value problems of first order is established independently of the grating period and the Fredholm indices for the oblique derivative and the classic Neumann boundary-value problems are given.