Geodesic
On convergence of the polynomial collocation method for singular integral equations and periodic pseudodifferential equations
Lobachevskii journal of mathematics, Tome 7 (2000), pp. 3-14

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We prove the convergence of polynomial collocation method for periodic singular integral, pseudodifferential and the systems of pseudodifferential equations in Sobolev spaces $H^s$ via the equivalence between the collocation and modified Galerkin methods. The boundness of the Lagrange interpolation operator in this spaces when $s>1/2$ allows to obtain the optimal error estimate for the approximate solution i.e. it has the same rate as the best approximation of the exact solution by the polynomials.
Keywords: singular integral equations, periodic pseudodifferentialequations, Galerkin method, collocation method. 
@article{LJM_2000_7_a0,
     author = {A. I. Fedotov},
     title = {On convergence of the polynomial collocation method for singular integral equations and periodic pseudodifferential equations},
     journal = {Lobachevskii journal of mathematics},
     pages = {3--14},
     publisher = {mathdoc},
     volume = {7},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_2000_7_a0/}
}
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A. I. Fedotov. On convergence of the polynomial collocation method for singular integral equations and periodic pseudodifferential equations. Lobachevskii journal of mathematics, Tome 7 (2000), pp. 3-14. http://geodesic.mathdoc.fr/item/LJM_2000_7_a0/