Finite sections for singular integral operators by weighted Chebyshev polynomials
The authors consider the equations (1) \(au + bS_\Gamma u = f\), \(au + S_\Gamma bu = f\) with piecewise continuous coefficients \(a,b\) on the interval \(\Gamma = [-1,1]\), with \(f \in L_2 (\Gamma)\) and with the Cauchy singular integral operator \(S_\Gamma \varphi (x) = \int_\Gamma (y - x)^{-1} \varphi (y)dy\). Equations (1) are replaced by the discrete analogues (2) \(P^\nu_n (aI + bS_\Gamma) P^\nu_nu = P^\nu_nf\), \(P^nu_n(aI + S_\Gamma b) P^\nu_nu = P^\nu_nf\), where \(I\) is the identity and \(P_n \{\sum^\infty_0 \xi_k u^\nu_k\} : = \sum^{n - 1}_0 \xi_k u^\nu_k\) are projectors on the \(n\)-dimensional subspaces generated by weighted trigonometric polynomials \(\{u^\nu_k\}^\infty_0\), \(\nu = 1,2,3,4\). The authors find conditions for the \(L_2\)-convergence of the solutions of the approximating equations (2) to the solution of the original equation (1) (i.e. the stability conditions). To formulate the main result of the paper let us recall the following notations from the paper: Denote by \(T = \{\zeta: |\zeta |= 1\}\) the unit circle then \(a^{(R)} (\zeta) : = a(Re \zeta)\) is a special lifting of a function \(a(x)\) from \(\Gamma\) to \(T\). Theorem: For piecewise-continuous coefficients \(a\) and \(b\) the finite section method (2) is stable if and only if the following holds: (i) \(A\) is an invertible operator; (ii) \([a^{(R)} I - b^{(R)}s] (I + S_T) + (I - S_T)\) is an invertible operator in \(L_2 (T)\); (iii) for every \(x \in (-1,1)\) the origin lies outside the triangle with vertices at the points \([a(x + 0) - b(x + 0)]/[a(x - 0) - b(x - 0)]\), \([a(x + 0) + b(x + 0)]/[a(x - 0) + b(x - 0)]\), and 1.
scientific article; zbMATH DE number 765286
scientific article