Abstract
We study optimal quadrature formulas for arbitrary weighted integrals and integrands from the Sobolev space \(H^1([0,1])\). We obtain general formulas for the worst case error depending on the nodes \(x_j\). A particular case is the computation of Fourier coefficients, where the oscillatory weight is given by \({\varrho }_k(x) = \exp (- 2 \pi i k x)\). Here we study the question whether equidistant nodes are optimal or not. We prove that this depends on n and k: equidistant nodes are optimal if \(n \ge 2.7 |k| +1 \) but might be suboptimal for small n. In particular, the equidistant nodes \(x_j = j/ |k|\) for \(j=0, 1, \ldots , |k| = n+1\) are the worst possible nodes and do not give any useful information.
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Acknowledgements
This work was started while S. Zhang was visiting Theoretical Numerical Analysis Group at Friedrich-Schiller-Universität Jena. He is extremely grateful for their hospitality. We thank both referees for their valuable remarks. One of them suggested a much shorter and more elegant proof of the results in Sect. 3.
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Dedicated to Henryk Woźniakowski on the occasion of his 70th birthday.
S. Zhang is partially supported by the National Natural Science Foundation of China (No. 11301002) and Natural Science Foundation for the Higher Education Institutions of Anhui Province of China (KJ2018A0017). E. Novak is partially supported by the DFG-Priority Program 1324.
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Zhang, S., Novak, E. Optimal Quadrature Formulas for the Sobolev Space \(H^1\). J Sci Comput 78, 274–289 (2019). https://doi.org/10.1007/s10915-018-0766-y
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DOI: https://doi.org/10.1007/s10915-018-0766-y