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In this paper, general real Kronrod extensions of Gaussian quadrature formulas with multiple nodes are introduced. A proof of their existence and uniqueness is given. In some cases, the explicit expressions of polynomials, whose zeros are the nodes of the considered quadratures, are determined. Very effective error bounds of the Gauss — Turán — Kronrod quadrature formulas, with Gori — Micchelli weight functions, for functions analytic on confocal ellipses, are derived.

Kronrod Extensions of Gaussian Quadratures with Multiple Nodes