Error estimates for an approximation to a characteristic singular integral equation (Q5951187)

The authors consider a characteristic singular integral equation \[ {\mathbf K}\varphi\equiv a(t)\varphi(t)+\frac{b(t)}{\pi i}\int_L\frac{\varphi(\tau) d\tau}{\tau-t}=f(t),\qquad t\in L:|z|=1, \tag{1} \] where \(a(t)\), \(b(t)\), and \(f(t)\) are given Hölder functions, and the unknown function \(\varphi(t)\) is assumed to be Hölder continuous. A rational operator of the de la Vallée-Poussin type of order \(4n\) is used to approximate the coefficients of (1). It is proved that the order of approximation of the exact solution of (1) with a nonnegative index of the equation in the space of continuous functions \(C(L)\) coincides with the order of approximation of the coefficients by rational functions. Suppose that the coefficients of (1) have absolutely continuous derivatives of order \(r-1\) and derivatives of order \(r\) are functions of bounded variation. Then, for an arbitrary value of the index, the approximate solution of problem (1) tends to an exact solution in the space \(C(L)\) at the rate \(\|\varphi(t)-\widetilde{\varphi}(t)\|_{C(L)}=O(n^{-(r+1)}).\)