{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T11:36:09Z","timestamp":1773920169430,"version":"3.50.1"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T00:00:00Z","timestamp":1766966400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T00:00:00Z","timestamp":1766966400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100013296","name":"Max Planck Institute for Mathematics in the Sciences","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100013296","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Various computational problems as, e.g., equations with fractional diffusion operators, evaluation of high-dimensional integrals, the M\u00f8ller\u2013Plesset approach in quantum chemistry, etc., are easily solved by using approximations by rational functions or by exponential sums. In the case of Cauchy\u2013Stieltjes or, respectively, Lebesgue\u2013Stieltjes 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.<\/jats:p>","DOI":"10.1007\/s00211-025-01523-1","type":"journal-article","created":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T11:33:39Z","timestamp":1767008019000},"page":"455-490","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the approximation of Stieltjes functions by exponential sums and rational functions with applications to partial differential equations"],"prefix":"10.1007","volume":"158","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-7232-0299","authenticated-orcid":false,"given":"Dietrich","family":"Braess","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4801-6189","authenticated-orcid":false,"given":"Wolfgang","family":"Hackbusch","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,12,29]]},"reference":[{"key":"1523_CR1","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/0009-2614(91)80078-C","volume":"176","author":"J Alml\u00f6f","year":"1991","unstructured":"Alml\u00f6f, J.: Elimination of energy denominators in M\u00f8ller\u2013Plesset perturbation theory by a Laplace transform approach. Chem. Phys. Lett. 176, 319\u2013320 (1991)","journal-title":"Chem. Phys. Lett."},{"key":"1523_CR2","doi-asserted-by":"publisher","first-page":"744","DOI":"10.1137\/140978223","volume":"54","author":"M Bachmayr","year":"2016","unstructured":"Bachmayr, M., Dahmen, W.: Adaptive low rank methods: problems on Sobolev spaces. SIAM J. Numer. Anal. 54, 744\u2013796 (2016)","journal-title":"SIAM J. Numer. Anal."},{"key":"1523_CR3","doi-asserted-by":"publisher","first-page":"419","DOI":"10.2140\/pjm.1960.10.419","volume":"10","author":"AV Balakrishnan","year":"1960","unstructured":"Balakrishnan, A.V.: Fractional powers of closed operators and the semigroups generated by them. Pac. J. Math. 10, 419\u2013437 (1960)","journal-title":"Pac. J. Math."},{"key":"1523_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00211-022-01329-5","volume":"153","author":"L Banjai","year":"2020","unstructured":"Banjai, L., Melenk, J.M., Schwab, C.: Exponential convergence of hp-FEM for spectral fractional Laplacian in polygons. Numer. Math. 153, 1\u201347 (2020)","journal-title":"Numer. Math."},{"key":"1523_CR5","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1007\/s11075-022-01256-4","volume":"91","author":"B Beckermann","year":"2022","unstructured":"Beckermann, B., Bisch, J., Luce, R.: On the rational approximation of Markov functions, with applications to the computation of Markov functions of Toeplitz matrices. Numer. Algorithms 91, 109\u2013144 (2022)","journal-title":"Numer. Algorithms"},{"key":"1523_CR6","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/j.acha.2009.08.011","volume":"28","author":"G Belkin","year":"2010","unstructured":"Belkin, G., Monz\u00f3n, L.: Approximation by exponential sums revisited. Appl. Comp. Harmonic Anal. 28, 131\u2013149 (2010)","journal-title":"Appl. Comp. Harmonic Anal."},{"key":"1523_CR7","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/s00791-018-0289-y","volume":"19","author":"A Bonito","year":"2018","unstructured":"Bonito, A., Borthagaray, J.P., Nochetto, R.H., Ot\u00e1rola, E., Salgado, A.J.: Numerical approximation for fractional diffusion. Comput. Vis. Sci. 19, 19\u201346 (2018)","journal-title":"Comput. Vis. Sci."},{"key":"1523_CR8","volume-title":"Pi and the AGM","author":"JM Borwein","year":"1987","unstructured":"Borwein, J.M., Borwein, P.B.: Pi and the AGM. Wiley, Hoboken (1987)"},{"key":"1523_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2020.113191","volume":"369","author":"W Boukaram","year":"2020","unstructured":"Boukaram, W., Lucchesi, M., Turkiyyah, G., Le Ma\u00eetre, O., Knio, O., Keyes, D.: Hierarchical matrix approximations for space-fractional diffusion equations. Comput. Methods Appl. Mech. Eng. 369, 113191 (2020)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1523_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61609-9","volume-title":"Nonlinear Approximation Theory","author":"D Braess","year":"1986","unstructured":"Braess, D.: Nonlinear Approximation Theory. Springer, Berlin (1986)"},{"key":"1523_CR11","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1006\/jath.1995.1110","volume":"83","author":"D Braess","year":"1995","unstructured":"Braess, D.: Asymptotics for the approximation of wave functions by exponential sums. J. Approx. Theory 83, 93\u2013103 (1995)","journal-title":"J. Approx. Theory"},{"key":"1523_CR12","doi-asserted-by":"crossref","unstructured":"Braess, D., Hackbusch, W.: On the efficient computation of high-dimensional integrals and the approximation by exponential sums. In: DeVore, R., Kunoth, A. (eds.), Multiscale, Nonlinear and Adaptive Approximation, pp. 39\u201374. Springer, Berlin (2009)","DOI":"10.1007\/978-3-642-03413-8_3"},{"key":"1523_CR13","doi-asserted-by":"crossref","unstructured":"Braess, D., Hackbusch, W.: The approximation of Cauchy\u2013Stieltjes and Laplace\u2013Stieltjes functions. In: DeVore, R., Kunoth, A. (eds.), Multiscale, Nonlinear and Adaptive Approximation II, pp. 115\u2013143. Springer, Berlin (2024)","DOI":"10.1007\/978-3-031-75802-7_7"},{"key":"1523_CR14","doi-asserted-by":"crossref","unstructured":"Canc\u00e8s, E., Defranceschi, M., Kutzelnigg, W., Le Bris, C., Maday, Y.: Computational quantum chemistry: a primer. In: Le Bris, C. (ed.), Handbook of Numerical Analysis, X, pp. 3\u2013270. Elsevier, Amsterdam (2003)","DOI":"10.1016\/S1570-8659(03)10003-8"},{"key":"1523_CR15","unstructured":"Danczul, T.: Order reduction for fractional diffusion problems. Doctoral thesis, Vienna (2021)"},{"issue":"5","key":"1523_CR16","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2488","volume":"30","author":"T Danczul","year":"2023","unstructured":"Danczul, T., Hofreiter, C., Sch\u00f6berl, J.: A unified rational Krylov method for elliptic and parabolic fractional diffusion problems. Numer. Linear Algebra Appl. 30(5), e2488 (2023)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"6","key":"1523_CR17","doi-asserted-by":"publisher","first-page":"2601","DOI":"10.1137\/22M152493X","volume":"61","author":"M Faustmann","year":"2023","unstructured":"Faustmann, M., Marcati, C., Melenk, J.M., Schwab, C.: Exponential convergence of hp-FEM for the integral fractional Laplacian in polygons. SIAM J. Numer. Anal. 61(6), 2601\u20132622 (2023)","journal-title":"SIAM J. Numer. Anal."},{"key":"1523_CR18","unstructured":"Ganelius, T., Hayman, W.K., Newman, D.J.: Lectures on approximation and value distribution. Presses de l\u2019Universit\u00e9 de Montr\u00e9al, Montr\u00e9al (1982)"},{"key":"1523_CR19","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1070\/SM1978v034n02ABEH001152","volume":"24","author":"AA Gon\u010dar","year":"1978","unstructured":"Gon\u010dar, A.A.: On the speed of rational approximation of some analytic functions. Math. USSR Sbornik 24, 131\u2013145 (1978)","journal-title":"Math. USSR Sbornik"},{"key":"1523_CR20","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1007\/s00607-003-0037-z","volume":"72","author":"L Grasedyck","year":"2004","unstructured":"Grasedyck, L.: Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure. Computing 72, 247\u2013265 (2004)","journal-title":"Computing"},{"key":"1523_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2023.115608","volume":"440","author":"G Li","year":"2024","unstructured":"Li, G.: Wavelet-based edge multiscale parareal algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain. J. Comput. Appl. Math. 440, 115608 (2024)","journal-title":"J. Comput. Appl. Math."},{"key":"1523_CR22","unstructured":"Hackbusch, W.: Webpages. www.mis.mpg.de\/scicomp\/EXP_SUM"},{"key":"1523_CR23","unstructured":"Hackbusch, W.: Webpages. www.mis.mpg.de\/scicomp\/RationalAppr"},{"key":"1523_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-47324-5","volume-title":"Hierarchical Matrices: Algorithms and Analysis","author":"W Hackbusch","year":"2015","unstructured":"Hackbusch, W.: Hierarchical Matrices: Algorithms and Analysis. Springer, Berlin (2015)"},{"key":"1523_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-35554-8","volume-title":"Tensor Spaces and Numerical Tensor Calculus","author":"W Hackbusch","year":"2019","unstructured":"Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus, 2nd edn. Springer, Berlin (2019)","edition":"2"},{"key":"1523_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00791-018-00308-4","volume":"20","author":"W Hackbusch","year":"2019","unstructured":"Hackbusch, W.: Computation of best $$L^{\\infty } $$ exponential sums for $$1\/x$$ by Remez\u2019 algorithm. Comput. Vis. Sci. 20, 1\u201311 (2019)","journal-title":"Comput. Vis. Sci."},{"key":"1523_CR27","volume-title":"Total Positivity","author":"S Karlin","year":"1968","unstructured":"Karlin, S.: Total Positivity. Stanford University Press, Stanford (1968)"},{"key":"1523_CR28","doi-asserted-by":"publisher","first-page":"1263","DOI":"10.1093\/imanum\/drac022","volume":"43","author":"U Khristenko","year":"2023","unstructured":"Khristenko, U., Wohlmuth, B.: Solving time-fractional differential equations via rational approximation. IMA J. Numer. Anal. 43, 1263\u20131290 (2023)","journal-title":"IMA J. Numer. Anal."},{"key":"1523_CR29","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1002\/qua.560510612","volume":"51","author":"W Kutzelnigg","year":"1994","unstructured":"Kutzelnigg, W.: Theory of the expansion of wave functions in a Gaussian basis. Int. J. Quantum Chem. 51, 447\u2013463 (1994)","journal-title":"Int. J. Quantum Chem."},{"key":"1523_CR30","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/BF01170633","volume":"38","author":"L L\u00f6wner","year":"1934","unstructured":"L\u00f6wner, L.: \u00dcber monotone Matrixfunktionen. Math. Z. 38, 177\u2013216 (1934)","journal-title":"Math. Z."},{"key":"1523_CR31","doi-asserted-by":"publisher","first-page":"A1494","DOI":"10.1137\/16M1106122","volume":"40","author":"Y Nakatsukasa","year":"2018","unstructured":"Nakatsukasa, Y., S\u00e8te, O., Trefethen, L.N.: The AAA algorithm for rational approximation. SIAM J. Sci. Comput. 40, A1494\u2013A1522 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"1523_CR32","doi-asserted-by":"publisher","first-page":"452","DOI":"10.1007\/BF01295091","volume":"67","author":"R Rutishauser","year":"1963","unstructured":"Rutishauser, R.: Betrachtungen zur Quadratwurzeliteration. Monatshefte Math. 67, 452\u2013464 (1963)","journal-title":"Monatshefte Math."},{"key":"1523_CR33","doi-asserted-by":"publisher","first-page":"841","DOI":"10.1007\/s00211-016-0856-4","volume":"136","author":"S Scholz","year":"2017","unstructured":"Scholz, S., Yserentant, H.: On the approximation of electronic wave-functions by anisotronic Gauss and Gauss\u2013Hermite functions. Numer. Math. 136, 841\u2013874 (2017)","journal-title":"Numer. Math."},{"key":"1523_CR34","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1007\/BF02392691","volume":"190","author":"HR Stahl","year":"2003","unstructured":"Stahl, H.R.: Best uniform rational approximation of $$x^{\\alpha }$$ on $$[0,1]$$. Acta Math. 190, 241\u2013306 (2003)","journal-title":"Acta Math."},{"key":"1523_CR35","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2706-9","volume-title":"Numerical Methods Based of Sinc and Analytic Functions","author":"F Stenger","year":"1993","unstructured":"Stenger, F.: Numerical Methods Based of Sinc and Analytic Functions. Springer, New York (1993)"},{"key":"1523_CR36","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/BF02145384","volume":"2","author":"RS Varga","year":"1992","unstructured":"Varga, R.S., Carpenter, A.J.: Some numerical results on best uniform rational approximation of $$x^{\\alpha }$$ on $$[0,1]$$. Numer. Algorithms 2, 171\u2013186 (1992)","journal-title":"Numer. Algorithms"},{"key":"1523_CR37","doi-asserted-by":"publisher","DOI":"10.1063\/1.2958921","volume":"129","author":"A Takatsuka","year":"2008","unstructured":"Takatsuka, A., Ten-no, S., Hackbusch, W.: Minimax approximation for the decomposition of energy denominators in Laplace-transformed M\u00f8ller\u2013Plesset perturbation theories. J. Chem. Phys. 129, 044112 (2008)","journal-title":"J. Chem. Phys."},{"key":"1523_CR38","doi-asserted-by":"publisher","first-page":"777","DOI":"10.1007\/s00211-024-01401-2","volume":"156","author":"H Yserentant","year":"2024","unstructured":"Yserentant, H.: An iterative method for the solution of Laplace-like equations in high and very high space dimensions. Numer. Math. 156, 777\u2013811 (2024)","journal-title":"Numer. Math."},{"key":"#cr-split#-1523_CR39.1","unstructured":"Zolotarev, E.I.: Application of elliptic functions to questions of functions deviating least and most from zero (Russian). Zap. Imp. Akad. Nauk (1877) (St. Petersburg 30 no. 5"},{"key":"#cr-split#-1523_CR39.2","unstructured":"reprinted in collected works II, pp. 1-59. Izdat, Akad. Nauk SSSR, Moscow (1932))"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01523-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-025-01523-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01523-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T06:48:05Z","timestamp":1773902885000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-025-01523-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,29]]},"references-count":40,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["1523"],"URL":"https:\/\/doi.org\/10.1007\/s00211-025-01523-1","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,29]]},"assertion":[{"value":"28 October 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 October 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 November 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 December 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}