{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T13:34:49Z","timestamp":1768743289080,"version":"3.49.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,11,5]],"date-time":"2019-11-05T00:00:00Z","timestamp":1572912000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,11,5]],"date-time":"2019-11-05T00:00:00Z","timestamp":1572912000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2020,10]]},"DOI":"10.1007\/s11075-019-00820-9","type":"journal-article","created":{"date-parts":[[2019,11,5]],"date-time":"2019-11-05T14:05:13Z","timestamp":1572962713000},"page":"427-448","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Multiscale radial kernels with high-order generalized Strang-Fix conditions"],"prefix":"10.1007","volume":"85","author":[{"given":"Wenwu","family":"Gao","sequence":"first","affiliation":[]},{"given":"Xuan","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,11,5]]},"reference":[{"key":"820_CR1","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1016\/j.cam.2011.07.002","volume":"236","author":"M Bozzini","year":"2011","unstructured":"Bozzini, M., Dyn, N., Rossini, M.: Construction of generators of quasi-interpolation operators of high approximation orders in spaces of polyharmonic splines. J. Comput. Appl. Math. 236, 577\u2013564 (2011)","journal-title":"J. Comput. Appl. Math."},{"key":"820_CR2","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s10444-012-9277-9","volume":"39","author":"M Bozzini","year":"2013","unstructured":"Bozzini, M., Rossini, M., Schaback, R.: Generalized Whittle-Mat\u00e9rn and polyharmonic kernels. Adv. Comput. Math. 39, 129\u2013142 (2013)","journal-title":"Adv. Comput. Math."},{"key":"820_CR3","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/s10444-014-9366-z","volume":"41","author":"M Bozzini","year":"2015","unstructured":"Bozzini, M., Rossini, M., Schaback, R., Volont\u00e8, E.: Radial kernels via scale derivatives. Adv. Comput. Math. 41, 277\u2013291 (2015)","journal-title":"Adv. Comput. Math."},{"key":"820_CR4","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1093\/imanum\/drt071","volume":"35","author":"M Bozzini","year":"2015","unstructured":"Bozzini, M., Lenarduzzi, L., Rossini, M., Schaback, R.: Interpolation with variably scaled kernels. IMA J. Numer. Anal. 35, 199\u2013219 (2015)","journal-title":"IMA J. Numer. Anal."},{"key":"820_CR5","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1017\/S0013091500019416","volume":"41","author":"M Buhmann","year":"1988","unstructured":"Buhmann, M.: Radial functions on compact support. Pocced. Edinburgh. Math. Soci. 41, 33\u201346 (1988)","journal-title":"Pocced. Edinburgh. Math. Soci."},{"key":"820_CR6","unstructured":"Buhmann, M.: Multivariable Interpolation Using Radial Basis Functions, Ph.D. dissertation, University of Cambridge (1989)"},{"key":"820_CR7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0962492900000015","volume":"1","author":"M Buhmann","year":"2000","unstructured":"Buhmann, M.: Radial basis functions. Acta Numer. 1, 1\u201338 (2000)","journal-title":"Acta Numer."},{"key":"820_CR8","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511543241","volume-title":"Radial Basis Functions: Theory and Implementations, Cambridge Monographs on Applied and Computational Mathematics","author":"M Buhmann","year":"2003","unstructured":"Buhmann, M.: Radial Basis Functions: Theory and Implementations, Cambridge Monographs on Applied and Computational Mathematics. Cambridge Univ. Press, Cambridge (2003)"},{"key":"820_CR9","first-page":"227","volume":"138","author":"E Cheney","year":"1992","unstructured":"Cheney, E., Light, W., Xu, Y.: On kernels and approximation orders, Approximation Theory (Memphis, TN). Lecture Notes in Pure and Applied Mathematics, Dekker, NewYork 138, 227\u2013242 (1992)","journal-title":"Lecture Notes in Pure and Applied Mathematics, Dekker, NewYork"},{"key":"820_CR10","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF02801574","volume":"78","author":"N Dyn","year":"1992","unstructured":"Dyn, N., Jackson, I., Levin, D., Ron, A.: On multivariate approximation by integer translates of a basis function. Israel J. Math. 78, 95\u2013130 (1992)","journal-title":"Israel J. Math."},{"key":"820_CR11","doi-asserted-by":"crossref","DOI":"10.1142\/6437","volume-title":"Meshfree Approximation Methods with MATLAB","author":"GE Fasshauer","year":"2007","unstructured":"Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB. World Scientific Publishers, Singapore (2007)"},{"key":"820_CR12","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1017\/S0962492914000130","volume":"24","author":"B Fornberg","year":"2015","unstructured":"Fornberg, B., Flyer, N.: Solving PDEs with radial basis functions. Acta Numer. 24, 215\u2013258 (2015)","journal-title":"Acta Numer."},{"key":"820_CR13","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1016\/j.camwa.2006.04.004","volume":"51","author":"B Fornberg","year":"2006","unstructured":"Fornberg, B., Larsson, E., Wright, G.: A new class of oscillatory radial basis functions. Comput. Math. Appl. 51, 1209\u20131222 (2006)","journal-title":"Comput. Math. Appl."},{"key":"820_CR14","doi-asserted-by":"crossref","first-page":"3256","DOI":"10.1016\/j.cam.2012.02.028","volume":"236","author":"WW Gao","year":"2012","unstructured":"Gao, W.W., Wu, Z.M.: Quasi-interpolation for linear functional data. J. Comput. Appl. Math. 236, 3256\u20133264 (2012)","journal-title":"J. Comput. Appl. Math."},{"key":"820_CR15","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.cam.2014.03.012","volume":"271","author":"WW Gao","year":"2014","unstructured":"Gao, W.W., Wu, Z.M.: A quasi-interpolation scheme for periodic data based on multiquadric trigonometric B-splines. J. Comput. Appl. Math. 271, 20\u201330 (2014)","journal-title":"J. Comput. Appl. Math."},{"key":"820_CR16","doi-asserted-by":"crossref","first-page":"1355","DOI":"10.1007\/s10444-017-9528-x","volume":"43","author":"WW Gao","year":"2017","unstructured":"Gao, W.W., Wu, Z.M.: Constructing radial kernels with higher-order generalized Strang-Fix conditions. Adv. Comput. Math. 43, 1355\u20131375 (2017)","journal-title":"Adv. Comput. Math."},{"key":"820_CR17","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s11075-017-0313-1","volume":"77","author":"WW Gao","year":"2018","unstructured":"Gao, W.W., Zhang, R.: Multiquadric trigonometric spline quasi-interpolation for numerical differentiation of noisy data: a stochastic perspective. Numer. Algor. 77, 243\u2013259 (2018)","journal-title":"Numer. Algor."},{"key":"820_CR18","unstructured":"Gao, W.W., Sun, X.P., Wu, Z.M., Zhou, X.: Multivariate Monte Carlo approximation based on scattered data, submitted"},{"key":"820_CR19","unstructured":"Gao, W.W., Fasshauer, G.E., Sun, X.P., Zhou, X.: Optimality and regularization properties of quasi-interpolation: deterministic and stochastic approaches, submitted"},{"key":"820_CR20","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1006\/jath.1993.1062","volume":"74","author":"RQ Jia","year":"1993","unstructured":"Jia, R.Q., Lei, J.J.: A new version of Strang-Fix conditions. J. Approx. Theory 74, 221\u2013225 (1993)","journal-title":"J. Approx. Theory"},{"key":"820_CR21","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.cam.2017.06.012","volume":"328","author":"MM Kazemi","year":"2018","unstructured":"Kazemi, M.M., Dehghan, M., Bastani, A.F.: On a new family of radial basis functions. J. Comput. A.pl. Math. 328, 75\u2013100 (2018)","journal-title":"J. Comput. A.pl. Math."},{"key":"820_CR22","first-page":"191","volume":"414","author":"V Maz\u2019ya","year":"2000","unstructured":"Maz\u2019ya, V., Schmidt, G.: Construction of basis functions for high order approximate approximations, Mathematical Aspects of boundary elements methods (Palaiseau, 1998), Chapman and hall\/CRC Res. Notes Math. 414, 191\u2013202 (2000)","journal-title":"Notes Math."},{"key":"820_CR23","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1006\/jath.2001.3556","volume":"110","author":"V Maz\u2019ya","year":"2001","unstructured":"Maz\u2019ya, V., Schmidt, G.: On quasi-interpolation with non-uniformly distributed centers on domains and manifolds. J. Approx. Theory 110, 125\u2013145 (2001)","journal-title":"J. Approx. Theory"},{"key":"820_CR24","first-page":"139","volume":"24","author":"C Rabut","year":"1991","unstructured":"Rabut, C.: An introduction to Schoenberg\u2019s approximation. Comput. Math. Appl. 24, 139\u2013175 (1991)","journal-title":"Comput. Math. Appl."},{"key":"820_CR25","volume-title":"Introduction to Fourier Analysis on Euclidean Spaces","author":"E Stein","year":"1971","unstructured":"Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press, Princeton (1971)"},{"key":"820_CR26","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s10444-009-9142-7","volume":"34","author":"R Schaback","year":"2011","unstructured":"Schaback, R.: The missing Wendland functions. Adv. Comp. Math. 34, 67\u201381 (2011)","journal-title":"Adv. Comp. Math."},{"key":"820_CR27","doi-asserted-by":"crossref","unstructured":"Strang, G., Fix, G.: A fourier analysis of the finite-element method. In: Geymonat, G. (ed.) Constructive Aspects of Functional Analysis, pp 793\u2013840. C.I.M.E., Rome (1973)","DOI":"10.1007\/978-3-642-10984-3_7"},{"key":"820_CR28","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/BF02123482","volume":"4","author":"H Wendland","year":"1995","unstructured":"Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389\u2013396 (1995)","journal-title":"Adv. Comput. Math."},{"key":"820_CR29","first-page":"1","volume":"8","author":"ZM Wu","year":"1992","unstructured":"Wu, Z.M.: Hermite-birkhoff interpolation of scattered data by radial basis functions. Approx. Theory. Appl. 8, 1\u201310 (1992)","journal-title":"Approx. Theory. Appl."},{"key":"820_CR30","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/BF03177517","volume":"4","author":"ZM Wu","year":"1995","unstructured":"Wu, Z.M.: Multivariate compactly supported positive definite radial basis functions. Adv. Comput. Math. 4, 283\u2013292 (1995)","journal-title":"Adv. Comput. Math."},{"key":"820_CR31","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1093\/imanum\/13.1.13","volume":"13","author":"ZM Wu","year":"1993","unstructured":"Wu, Z.M., Schaback, R.: Local error estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13, 13\u201327 (1993)","journal-title":"IMA J. Numer. Anal."},{"key":"820_CR32","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/0377-0427(96)00047-7","volume":"73","author":"ZM Wu","year":"1996","unstructured":"Wu, Z.M.: Operators on radial functions. J. Comput. Appl. Math. 73, 257\u2013270 (1996)","journal-title":"J. Comput. Appl. Math."},{"key":"820_CR33","first-page":"115","volume":"84","author":"ZM Wu","year":"1997","unstructured":"Wu, Z.M.: Compactly supported positive definite radial basis functions and the Strang-Fix condition. Appl. Math Comput. 84, 115\u2013124 (1997)","journal-title":"Appl. Math Comput."},{"key":"820_CR34","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s10444-004-1832-6","volume":"23","author":"ZM Wu","year":"2005","unstructured":"Wu, Z.M., Liu, J.P.: Generalized Strang-Fix condition for scattered data quasi-interpolation. Adv. Comput. Math. 23, 201\u2013214 (2005)","journal-title":"Adv. Comput. Math."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00820-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-019-00820-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00820-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,11,4]],"date-time":"2020-11-04T00:17:29Z","timestamp":1604449049000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-019-00820-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,5]]},"references-count":34,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,10]]}},"alternative-id":["820"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00820-9","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,11,5]]},"assertion":[{"value":"8 June 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 September 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 November 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}