{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T00:56:14Z","timestamp":1772499374294,"version":"3.50.1"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,7,10]],"date-time":"2020-07-10T00:00:00Z","timestamp":1594339200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,7,10]],"date-time":"2020-07-10T00:00:00Z","timestamp":1594339200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Japan Society of the Promotion of Science with KAKENHI","award":["17K14241."],"award-info":[{"award-number":["17K14241."]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2021,5]]},"DOI":"10.1007\/s11075-020-00973-y","type":"journal-article","created":{"date-parts":[[2020,7,10]],"date-time":"2020-07-10T01:03:34Z","timestamp":1594343014000},"page":"445-468","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Kernel-based interpolation at approximate Fekete points"],"prefix":"10.1007","volume":"87","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5984-7295","authenticated-orcid":false,"given":"Toni","family":"Karvonen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Simo","family":"S\u00e4rkk\u00e4","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ken\u2019ichiro","family":"Tanaka","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,7,10]]},"reference":[{"issue":"2","key":"973_CR1","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1007\/s00211-007-0092-z","volume":"107","author":"R Arcang\u00e9li","year":"2007","unstructured":"Arcang\u00e9li, R., de Silanes, M. C. L., Torrens, J. J.: An extension of a bound for functions in Sobolev spaces, with applications to (m,s)-spline interpolation and smoothing. Numer. Math. 107(2), 181\u2013211 (2007)","journal-title":"Numer. Math."},{"issue":"3","key":"973_CR2","doi-asserted-by":"publisher","first-page":"512","DOI":"10.1016\/j.jat.2009.08.004","volume":"162","author":"R Beatson","year":"2010","unstructured":"Beatson, R.: Error bounds for anisotropic RBF interpolation. J. Approx. Theory 162(3), 512\u2013527 (2010)","journal-title":"J. Approx. Theory"},{"key":"973_CR3","first-page":"12907","volume":"32","author":"A Belhadji","year":"2019","unstructured":"Belhadji, A., Bardenet, R., Chainais, P.: Kernel quadrature with DPPs. Adv. Neural Inf. Process. Syst. 32, 12907\u201312917 (2019)","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"973_CR4","doi-asserted-by":"crossref","unstructured":"Berlinet, A., Thomas-agnan, C.: Reproducing Kernel Hilbert Spaces in Probability and Statistics. Springer, Berlin (2004)","DOI":"10.1007\/978-1-4419-9096-9"},{"key":"973_CR5","first-page":"8","volume":"4","author":"L Bos","year":"2011","unstructured":"Bos, L., De Marchi, S.: On optimal points for interpolation by univariate exponential functions. Dol. Res. Notes Approx. 4, 8\u201312 (2011)","journal-title":"Dol. Res. Notes Approx."},{"issue":"5","key":"973_CR6","doi-asserted-by":"publisher","first-page":"1984","DOI":"10.1137\/090779024","volume":"48","author":"L Bos","year":"2010","unstructured":"Bos, L., De Marchi, S., Sommariva, A., Vianello, M.: Computing multivariate Fekete and Leja points by numerical linear algebra. SIAM J. Numer. Anal. 48(5), 1984\u20131999 (2010)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"973_CR7","doi-asserted-by":"publisher","first-page":"252","DOI":"10.1006\/jath.2002.3731","volume":"119","author":"LP Bos","year":"2002","unstructured":"Bos, L. P., Maier, U.: On the asymptotics of Fekete-type points for univariate radial basis interpolation. J. Approx. Theory 119(2), 252\u2013270 (2002)","journal-title":"J. Approx. Theory"},{"issue":"9","key":"973_CR8","doi-asserted-by":"publisher","first-page":"2477","DOI":"10.1016\/j.cam.2011.12.006","volume":"236","author":"M Briani","year":"2012","unstructured":"Briani, M., Sommariva, A., Vianello, M.: Computing Fekete and Lebesgue points: simplex, square, disk. J. Comput. Appl. Math. 236(9), 2477\u20132486 (2012)","journal-title":"J. Comput. Appl. Math."},{"key":"973_CR9","first-page":"16","volume":"2","author":"S De Marchi","year":"2009","unstructured":"De Marchi, S., Schaback, R.: Nonstandard kernels and their applications. Dol. Res. Notes Approx. 2, 16\u201343 (2009)","journal-title":"Dol. Res. Notes Approx."},{"key":"973_CR10","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1007\/s10444-008-9093-4","volume":"32","author":"S De Marchi","year":"2010","unstructured":"De Marchi, S., Schaback, R.: Stability of kernel-based interpolation. Adv. Comput. Math. 32, 155\u2013161 (2010)","journal-title":"Adv. Comput. Math."},{"issue":"3","key":"973_CR11","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1007\/s10444-004-1829-1","volume":"23","author":"S De Marchi","year":"2005","unstructured":"De Marchi, S., Schaback, R., Wendland, H.: Near-optimal data-independent point locations for radial basis function interpolation. Adv. Comput. Math. 23(3), 317\u2013330 (2005)","journal-title":"Adv. Comput. Math."},{"issue":"1","key":"973_CR12","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1137\/10080138X","volume":"50","author":"G Fasshauer","year":"2012","unstructured":"Fasshauer, G., Hickernell, F., Wo\u017aniakowski, H.: On dimension-independent rates of convergence for function approximation with Gaussian kernels. SIAM J. Numer. Anal. 50(1), 247\u2013271 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"973_CR13","doi-asserted-by":"crossref","unstructured":"Fasshauer, G., McCourt, M.: Kernel-based Approximation Methods Using MATLAB. Number 19 in Interdisciplinary Mathematical Sciences. World Scientific Publishing (2015)","DOI":"10.1142\/9335"},{"key":"973_CR14","doi-asserted-by":"crossref","unstructured":"Fasshauer, G. E.: Meshfree Approximation Methods with MATLAB. Number 6 in Interdisciplinary Mathematical Sciences. World Scientific Publishing (2007)","DOI":"10.1142\/6437"},{"issue":"2","key":"973_CR15","doi-asserted-by":"publisher","first-page":"A737","DOI":"10.1137\/110824784","volume":"34","author":"GE Fasshauer","year":"2012","unstructured":"Fasshauer, G. E., McCourt, M. J.: Stable evaluation of Gaussian radial basis function interpolants. SIAM J. Sci. Comput. 34(2), A737\u2013A762 (2012)","journal-title":"SIAM J. Sci. Comput."},{"key":"973_CR16","first-page":"7768","volume":"32","author":"G Gautier","year":"2019","unstructured":"Gautier, G., Bardenet, R., Valko, M.: On two ways to use determinantal point processes for Monte Carlo integration. Adv. Neural Inf. Process. Syst. 32, 7768\u20137777 (2019)","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"973_CR17","unstructured":"Johansson, F., et al.: mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 1.10). http:\/\/mpmath.org\/ (2018)"},{"key":"973_CR18","doi-asserted-by":"publisher","unstructured":"Karvonen, T., S\u00e4rkk\u00e4, S.: Worst-case optimal approximation with increasingly flat Gaussian kernels. Advances in Computational Mathematics. Published online https:\/\/doi.org\/10.1007\/s10444-020-09767-1 (2020)","DOI":"10.1007\/s10444-020-09767-1"},{"issue":"2","key":"973_CR19","doi-asserted-by":"publisher","first-page":"537","DOI":"10.1137\/050642113","volume":"39","author":"YJ Lee","year":"2007","unstructured":"Lee, Y. J., Yoon, G. J., Yoon, J.: Convergence of increasingly flat radial basis interpolants to polynomial interpolants. SIAM J. Math. Anal. 39 (2), 537\u2013553 (2007)","journal-title":"SIAM J. Math. Anal."},{"issue":"2","key":"973_CR20","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1007\/s00365-009-9080-0","volume":"32","author":"HQ Minh","year":"2010","unstructured":"Minh, H. Q.: Some properties of Gaussian reproducing kernel Hilbert spaces and their implications for function approximation and learning theory. Constr. Approx. 32(2), 307\u2013338 (2010)","journal-title":"Constr. Approx."},{"key":"973_CR21","unstructured":"M\u00fcller, S.: Komplexit\u00e4t und Stabilit\u00e4t von kernbasierten Rekonstruktionsmethoden. PhD Thesis, Institut f\u00fcr Numerische und Angewandte Mathematik, Georg-August-Universit\u00e4t G\u00f6ttingen (2009)"},{"issue":"2","key":"973_CR22","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/s00365-005-0624-7","volume":"24","author":"FJ Narcowich","year":"2006","unstructured":"Narcowich, F. J., Ward, J. D., Wendland, H.: Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions. Constr. Approx. 24(2), 175\u2013186 (2006)","journal-title":"Constr. Approx."},{"key":"973_CR23","doi-asserted-by":"crossref","unstructured":"Paulsen, V. I., Raghupathi, M.: An Introduction to the Theory of Reproducing Kernel Hilbert Spaces. Number 152 in Cambridge Studies in Advanced Mathematics. Cambridge University Press (2016)","DOI":"10.1017\/CBO9781316219232"},{"key":"973_CR24","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/s10444-008-9089-0","volume":"32","author":"C Rieger","year":"2010","unstructured":"Rieger, C., Zwicknagl, B.: Sampling inequalities for infinitely smooth functions, with applications to interpolation and machine learning. Adv. Comput. Math. 32, 103\u2013129 (2010)","journal-title":"Adv. Comput. Math."},{"issue":"2","key":"973_CR25","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1007\/s00365-013-9211-5","volume":"39","author":"C Rieger","year":"2014","unstructured":"Rieger, C., Zwicknagl, B.: Improved exponential convergence rates by oversampling near the boundary. Constr. Approx. 39(2), 323\u2013341 (2014)","journal-title":"Constr. Approx."},{"issue":"1","key":"973_CR26","first-page":"26","volume":"62","author":"H Robbins","year":"1955","unstructured":"Robbins, H.: A remark on Stirling\u2019s formula. Am. Math. Mon. 62(1), 26\u201329 (1955)","journal-title":"Am. Math. Mon."},{"key":"973_CR27","first-page":"68","volume":"10","author":"G Santin","year":"2017","unstructured":"Santin, G., Haasdonk, B.: Convergence rate of the data-independent P-greedy algorithm in kernel-based approximation. Dol. Res. Notes Approx. 10, 68\u201378 (2017)","journal-title":"Dol. Res. Notes Approx."},{"issue":"225","key":"973_CR28","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1090\/S0025-5718-99-01009-1","volume":"68","author":"R Schaback","year":"1999","unstructured":"Schaback, R.: Improved error bounds for scattered data interpolation by radial basis functions. Math. Comput. 68(225), 201\u2013216 (1999)","journal-title":"Math. Comput."},{"issue":"1\u20132","key":"973_CR29","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/S0377-0427(00)00345-9","volume":"121","author":"R Schaback","year":"2000","unstructured":"Schaback, R.: A unified theory of radial basis functions: native Hilbert spaces for radial basis functions II. J. Comput. Appl. Math. 121(1\u20132), 165\u2013177 (2000)","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"973_CR30","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s00365-004-0585-2","volume":"21","author":"R Schaback","year":"2005","unstructured":"Schaback, R.: Multivariate interpolation by polynomials and radial basis functions. Constr. Approx. 21(3), 293\u2013317 (2005)","journal-title":"Constr. Approx."},{"key":"973_CR31","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jat.2018.05.002","volume":"235","author":"R Schaback","year":"2018","unstructured":"Schaback, R.: Superconvergence of kernel-based interpolation. J. Approx. Theory 235, 1\u201319 (2018)","journal-title":"J. Approx. Theory"},{"issue":"3","key":"973_CR32","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1023\/A:1019105612985","volume":"24","author":"R Schaback","year":"2000","unstructured":"Schaback, R., Wendland, H.: Adaptive greedy techniques for approximate solution of large RBF systems. Numer. Algorithm. 24(3), 239\u2013254 (2000)","journal-title":"Numer. Algorithm."},{"key":"973_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jco.2017.11.001","volume":"45","author":"IH Sloan","year":"2018","unstructured":"Sloan, I. H., Wo\u017aniakowski, H.: Multivariate approximation for analytic functions with Gaussian kernels. J. Complex. 45, 1\u201321 (2018)","journal-title":"J. Complex."},{"issue":"10","key":"973_CR34","doi-asserted-by":"publisher","first-page":"4635","DOI":"10.1109\/TIT.2006.881713","volume":"52","author":"I Steinwart","year":"2006","unstructured":"Steinwart, I., Hush, D., Scovel, C.: An explicit description of the reproducing kernel Hilbert spaces of Gaussian RBF kernels. IEEE Trans. Inf. Theory 52(10), 4635\u20134643 (2006)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"973_CR35","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1016\/j.jco.2004.09.002","volume":"21","author":"H Sun","year":"2005","unstructured":"Sun, H.: Mercer theorem for RKHS on noncompact sets. J. Complex. 21(3), 337\u2013349 (2005)","journal-title":"J. Complex."},{"key":"973_CR36","doi-asserted-by":"publisher","first-page":"1049","DOI":"10.1007\/s11075-019-00792-w","volume":"84","author":"K Tanaka","year":"2020","unstructured":"Tanaka, K.: Generation of point sets by convex optimization for interpolation in reproducing kernel Hilbert spaces. Numer. Algorithm. 84, 1049\u20131079 (2020)","journal-title":"Numer. Algorithm."},{"issue":"4","key":"973_CR37","doi-asserted-by":"publisher","first-page":"1957","DOI":"10.1093\/imanum\/dry056","volume":"39","author":"K Tanaka","year":"2019","unstructured":"Tanaka, K., Sugihara, M.: Design of accurate formulas for approximating functions in weighted Hardy spaces by discrete energy minimization. IMA J. Numer. Anal. 39(4), 1957\u20131984 (2019)","journal-title":"IMA J. Numer. Anal."},{"key":"973_CR38","unstructured":"Wendland, H.: Scattered Data Approximation. Number 17 in Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press (2005)"},{"issue":"4","key":"973_CR39","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1007\/s00211-005-0637-y","volume":"101","author":"H Wendland","year":"2005","unstructured":"Wendland, H., Rieger, C.: Approximate interpolation with applications to selecting smoothing parameters. Numer. Math. 101(4), 729\u2013748 (2005)","journal-title":"Numer. Math."},{"key":"973_CR40","first-page":"83","volume":"6","author":"D Wirtz","year":"2013","unstructured":"Wirtz, D., Haasdonk, B.: A vectorial kernel orthogonal greedy algorithm. Dol. Res. Notes Approx. 6, 83\u2013100 (2013)","journal-title":"Dol. Res. Notes Approx."}],"updated-by":[{"DOI":"10.1007\/s11075-020-01034-0","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2020,12,5]],"date-time":"2020-12-05T00:00:00Z","timestamp":1607126400000}}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00973-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-020-00973-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00973-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,31]],"date-time":"2021-07-31T00:44:14Z","timestamp":1627692254000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-020-00973-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,10]]},"references-count":40,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,5]]}},"alternative-id":["973"],"URL":"https:\/\/doi.org\/10.1007\/s11075-020-00973-y","relation":{"correction":[{"id-type":"doi","id":"10.1007\/s11075-020-01034-0","asserted-by":"object"}]},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,10]]},"assertion":[{"value":"16 December 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 June 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 July 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 December 2020","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s11075-020-01034-0","URL":"https:\/\/doi.org\/10.1007\/s11075-020-01034-0","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}]}}