A. I. Fedotov
On convergence of the polynomial collocation method 
for singular integral equations and periodic pseudodifferential equations

(Lobachevskii Journal of Mathematics, Vol.7, pp.3-14 )

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.

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