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<p>Various computational problems as, e.g., equations with fractional diffusion operators, evaluation of high-dimensional integrals, the Møller–Plesset approach in quantum chemistry, etc., are easily solved by using approximations by rational functions or by exponential sums. In the case of Cauchy–Stieltjes or, respectively, Lebesgue–Stieltjes functions we provide a uniform proof of upper bounds of the convergence rates of their best approximations by rational functions or exponential sums. It turns out that the convergence rate by rational approximation is better than for exponential sums. We extend the analysis also to the approximation on infinite intervals and to the best approximation of the relative error. Instead of looking for the best approximation one can use the computationally cheaper quadrature method, in particular the sinc quadrature. The corresponding sharp error estimates are determined. The theoretical results are supported by numerical results.</p>

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On the approximation of Stieltjes functions by exponential sums and rational functions with applications to partial differential equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月29日

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  • Dietrich Braess,
  • Wolfgang Hackbusch

摘要

Various computational problems as, e.g., equations with fractional diffusion operators, evaluation of high-dimensional integrals, the Møller–Plesset approach in quantum chemistry, etc., are easily solved by using approximations by rational functions or by exponential sums. In the case of Cauchy–Stieltjes or, respectively, Lebesgue–Stieltjes functions we provide a uniform proof of upper bounds of the convergence rates of their best approximations by rational functions or exponential sums. It turns out that the convergence rate by rational approximation is better than for exponential sums. We extend the analysis also to the approximation on infinite intervals and to the best approximation of the relative error. Instead of looking for the best approximation one can use the computationally cheaper quadrature method, in particular the sinc quadrature. The corresponding sharp error estimates are determined. The theoretical results are supported by numerical results.