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Asymptotically optimal quadrature rules for uniform splines over the real line

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Abstract

We provide explicit asymptotically optimal quadrature rules for uniform Ck-splines, k = 0,1, over the real line. The nodes of these quadrature rules are given in terms of the zeros of ultraspherical polynomials (Gegenbauer polynomials) and related polynomials. We conjecture that our derived rules are the only possible periodic asymptotically optimal quadrature rules for these spline spaces.

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Notes

  1. All the congruences given in this work can be verified by straightforward computation.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia

    Rachid Ait-Haddou

  2. Santa Fé, Cuba

    Helmut Ruhland

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  1. Rachid Ait-Haddou
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  2. Helmut Ruhland
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Correspondence to Rachid Ait-Haddou.

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Ait-Haddou, R., Ruhland, H. Asymptotically optimal quadrature rules for uniform splines over the real line. Numer Algor 86, 1189–1223 (2021). https://doi.org/10.1007/s11075-020-00929-2

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