Calcolo

, 55:10 | Cite as

On the dense unbounded divergence of interpolatory product integration on Jacobi nodes

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Abstract

This paper deals with interpolatory product integration rules based on Jacobi nodes, associated with the Banach space of all s-times continuously differentiable functions, and with a Banach space of absolutely integrable functions, on the interval \([-1,1]\) of the real axis. In order to highlight the topological structure of the set of unbounded divergence for the corresponding product quadrature formulas, a family of continuous linear operators associated with these product integration procedures is pointed out, and the unboundedness of the set of their norms is established, by means of some properties involving the theory of Jacobi polynomials. The main result of the paper is based on some principles of Functional Analysis, and emphasizes the phenomenon of double condensation of singularities with respect to the considered interpolatory product quadrature formulas, by pointing out large subsets (in topological meaning) of the considered Banach spaces, on which the quadrature procedures are unboundedly divergent.

Keywords

Product integration Jacobi nodes Superdense set Condensation of singularities 

Mathematics Subject Classification

41A10 65D32 

Notes

Acknowledgements

The author is grateful to the anonymous reviewers for their suggestions.

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Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTechnical UniversityCluj-NapocaRomania

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