ON CONNECTION OF ONE CLASS OF ONE-DIMENSIONAL PSEUDODIFFERENTIAL OPERATORS WITH SINGULAR INTEGRAL OPERATORS

Abstract

The paper discusses a homogeneous one-dimensional pseudodifferential equation with a symbol of the form

$$A(x, \xi)=A_0(\xi)+\sum\limits_{k=1}^N th\frac{\pi}{\alpha}\left (x-\lambda_k+i\frac{\alpha\beta}{2}\right )A_k(\xi)$$

$$(x, \xi, \lambda \in \mathbb{R},  \alpha>0,  -1<\beta<1,  k=1,2,…,N ), $$

where  $A_k(\xi)  ( k=0,1,...,N)$  are locally integrable functions from class of symbols of non-negative order r . The method of bringing the pseudodifferential equation to a system of onedimensional singular integral equations with Cauchy’s kernel is proposed.