Periodic Riemann problem and discrete convolution equations

On solvability of some difference-discrete equations

“…It was shown in [16,18] that solvability of the problem (3.2) or equivalently equation (3.1) is defined by the so called index of factorization [2-4, 7, 9, 16, 17]. We will give here the needed definition assuming everywhere that…”

“…It was done in papers [16,18], and here we will briefly describe these constructions. Let us introduce the following operators which are generated by the periodic analogue of the Hilbert transform, ξ = (ξ , ξ m ),…”

“…(3.1) The equation (3.1) is closely related to the so called periodic Riemann problem [18]. Let us denote Π ± the upper and lower half-strips in a complex plane C,…”

“…According to [18] there are many solutions for the equation (2.3). We will write these solutions taking into account that arbitrary constants will really depend on ξ .…”

“…Such equations [5,8,12,[14][15][16][17][18] arise in many applied problems, for example in control theory [6] and digital signal processing [1], thus the problem of their solvability is very topical. We choose the space L 2 as an initial functional space, but these equations can be considered in more general spaces L p .…”