Convolution Equations on a Large Finite Interval with Symbols Having Power-Order Zeros

Abstract

A certain convolution equation is studied on a large finite interval. This equation arose in acoustics for description of a wave conductor surface with a bed of ice. The main feature of this equation is that the symbol of the corresponding operator has zeros of power order in the dual variable, so that the inverse operator is a long-range one. A complete power-order asymptotic expansion is constructed for the kernel of the inverse operator as the length of the interval tends to infinity.