Geodesic
Justifying the convergence of the rectangular method for complete singular integral equations with continuous coefficients on the circle
Matematičeskie zametki, Tome 77 (2005) no. 2, pp. 163-175.

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For an integral equation on the unit circle $\Gamma$ of the form $(aI+bS+K)f=g$, where $a$ and $b$ are Hölder functions, $S$ is a singular integration operator, and $K$ is an integral operator with Hölder kernel, we consider a method of solution based on the discretization of integral operators using the rectangle rule. This method is justified under the assumption that the equation is solvable in $L_2(\Gamma)$ and the coefficients $a$ and $b$ satisfy the strong ellipticity condition. 
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M. É. Abramyan. Justifying the convergence of the rectangular method for complete singular integral equations with continuous coefficients on the circle. Matematičeskie zametki, Tome 77 (2005) no. 2, pp. 163-175. http://geodesic.mathdoc.fr/item/MZM_2005_77_2_a0/

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