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.
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References
Brass, H., Petras, K.: Quadrature Theorie. The Theory of Numerical Integration on a Compact Interval. American Mathematical Society, Providence (2011)
Cheney, E.W., Light, W.A.: A Course on Approximation Theory. American Mathematical Society, Providence (2009)
Cobzaş, S., Muntean, I.: Condensation of singularities and divergence results in approximation theory. J. Approx. Theory 31, 138–153 (1981)
Cvetković, A.S., Spalevic, M.M.: Estimating the error of Gauss-Turán quadrature formulas using their extensions. Electron. Trans. Numer. Anal. 41, 1–12 (2014)
Ehrich, S.: On product integration with Gauss-Kronrod nodes. SIAM J. Numer. Anal. 35, 78–92 (1998)
Folland, G.: Real Analysis. Modern Techniques and Applications. John and Willer Sons Inc, Hoboken (1999)
Milovanović, G.V., Spalević, M.M.: Gauss-Turán quadratures of Kronrod type for generalized Chebyshev weight functions. Calcolo 43, 171–195 (2006)
Mitrea, A.I.: Double condensation of singularities for product quadrature formulas with differentiable functions. Carpatian J. Math. 28, 83–91 (2012)
Mitrea, A.I.: On the dense divergence of the product quadrature formulas of interpolatory type. J. Math. Anal. Appl. 433, 1409–1414 (2016)
Nevai, G.P.: Mean convergence of Lagrange interpolation, I. J. Approx. Theory 18, 363–377 (1976)
Petrova, G.: Generalized Gauss-Radau and Gauss-Lobatto formulas with Jacobi weight functions. BIT Numer. Math. (2016). https://doi.org/10.1007/s10543-016-0627-8
Rabinowitz, P.: The convergence of interpolatory product integration rules. BIT 26, 131–134 (1986)
Rabinowitz, P.: On the convergence of interpolatory product integration rules based on Gauss, Radau and Lobatto points. Israel J. Math. 56, 66–74 (1986)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGrawll-Hill, New York (1987)
Sloan, I.H., Smith, W.E.: Properties of interpolatory product integration rules. SIAM J. Numer. Anal. 19, 427–442 (1982)
Smith, W.E., Paget, D.: Optimal nodes for interpolatory product integration. SIAM J. Numer. Anal. 29, 586–600 (1992)
Szabados, J., Vértesi, P.: Interpolation of Functions. World Sci. Publ. Co., Singapore (1990)
Szegö, G.: Orthogonal Polynomials. American Mathematical Society, Providence (1975)
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The author is grateful to the anonymous reviewers for their suggestions.
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Mitrea, A.I. On the dense unbounded divergence of interpolatory product integration on Jacobi nodes. Calcolo 55, 10 (2018). https://doi.org/10.1007/s10092-018-0253-8
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DOI: https://doi.org/10.1007/s10092-018-0253-8