Approximate solution of singular integral equations with Carleman shift in nontrivial kernel case (Q2703327)

The paper deals with the singular integral equation on the unit circle \(\gamma= \{t\in C:|t|=1\}\): NEWLINE\[NEWLINE \sum_{k=0}^{n-1}\Biggl\{\alpha_{k}(t)\phi[\alpha_{k}(t)]+ {c_{k}(t)\over\pi i}\int_{\gamma}{\phi(\tau)\over \tau- \alpha_{k}(t)} d\tau\Biggr\}+ \int_{\gamma}K(t,\tau)\phi(\tau) d\tau= g(t), NEWLINE\]NEWLINE where shift \(\alpha(t)\) satisfies the conditions \(\alpha_{m}(t)\equiv t, \alpha_{k}(t)=\alpha[\alpha_{k-1}(t)], \alpha_{0}(t)=t\), and \(m\geq 2\) is a minimal integer such that this condition holds true. The author justifies projecton methods for the approximate solution of the considered equation in the case when the homogeneous equation has nontrivial solutions.