Geodesic
Quadrature processes for integrals of Cauchy type
Matematičeskie zametki, Tome 11 (1972) no. 5, pp. 517-526.

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We study questions relating to convergence of the process $$ \int_{-1}^{+1}\rho(t)\frac{f(t)}{t-x}dt\approx\sum_{k=1}^n\alpha_{k,n}(x)f(x_k^{(n)})\qquad(-11), $$ wherein the singular integral is taken in the principal value sense. General conditions for convergence in the class of continuously differentiable functions $f$ are formulated. In the case of the weight function $\rho(t)=(\sqrt{1-t^2})^{-1}$, we investigate, under various assumptions on $f$, the convergence of a specific quadrature process. 
@article{MZM_1972_11_5_a5,
     author = {D. G. Sanikidze},
     title = {Quadrature processes for integrals of {Cauchy} type},
     journal = {Matemati\v{c}eskie zametki},
     pages = {517--526},
     publisher = {mathdoc},
     volume = {11},
     number = {5},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_11_5_a5/}
}
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D. G. Sanikidze. Quadrature processes for integrals of Cauchy type. Matematičeskie zametki, Tome 11 (1972) no. 5, pp. 517-526. http://geodesic.mathdoc.fr/item/MZM_1972_11_5_a5/