{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T14:42:22Z","timestamp":1775054542102,"version":"3.50.1"},"reference-count":49,"publisher":"Elsevier BV","issue":"1","license":[{"start":{"date-parts":[[2003,1,1]],"date-time":"2003-01-01T00:00:00Z","timestamp":1041379200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2003,1,1]],"date-time":"2003-01-01T00:00:00Z","timestamp":1041379200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2013,7,17]],"date-time":"2013-07-17T00:00:00Z","timestamp":1374019200000},"content-version":"vor","delay-in-days":3850,"URL":"http:\/\/www.elsevier.com\/open-access\/userlicense\/1.0\/"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Approximation Theory"],"published-print":{"date-parts":[[2003,1]]},"DOI":"10.1016\/s0021-9045(02)00011-4","type":"journal-article","created":{"date-parts":[[2003,1,21]],"date-time":"2003-01-21T13:45:29Z","timestamp":1043156729000},"page":"36-70","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":16,"title":["On best approximation of classes by radial functions"],"prefix":"10.1016","volume":"120","author":[{"given":"Vitaly","family":"Maiorov","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","reference":[{"key":"10.1016\/S0021-9045(02)00011-4_BIB1","series-title":"Sobolev Spaces","author":"Adams","year":"1975"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB2","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/BF02921656","article-title":"Approximation by spherical waves in Lp-space","volume":"6","author":"Agranovsky","year":"1996","journal-title":"J. Geom. Anal."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB3","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1155\/S1073792894000504","article-title":"Injectivity sets for the Radon transform over circles and complete systems of radial functions","volume":"11","author":"Agranovsky","year":"1994","journal-title":"Intern. Math. Res. Notes"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB4","doi-asserted-by":"crossref","first-page":"930","DOI":"10.1109\/18.256500","article-title":"Universal approximation bounds for superposition of a sigmoidal function","volume":"39","author":"Barron","year":"1993","journal-title":"IEEE Trans. Inform. Theory"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB5","series-title":"Interpolation of Operators","author":"Bennett","year":"1988"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB6","unstructured":"A. Bejancu, On the accuracy of surface spline approximation and interpolation to bump functions, DAMTP Technical Report, University of Cambridge, 2000."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB7","first-page":"129","article-title":"On a recovery problem","volume":"4","author":"Buhmann","year":"1997","journal-title":"Ann. Numer. Math."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB8","unstructured":"M. Buhmann (Ed.), Radial Basis Functions and Their Applications, Advanced in Computational Mathematics, Vol. 11, 1999."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB9","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1006\/aama.1998.0623","article-title":"Identifying linear combinations of Ridge functions","volume":"22","author":"Buhmann","year":"1999","journal-title":"Adv. Appl. Math."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0962492900000015","article-title":"Radial basis functions","volume":"9","author":"Buhman","year":"2000","journal-title":"Acta Numer."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB11","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/BF01203417","article-title":"On quasi-interpolation by radial basis functions with scattered centres","volume":"11","author":"Bumann","year":"1995","journal-title":"Constr. Approx."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB12","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1007\/BF01171759","article-title":"Optimal nonlinear approximation","volume":"63","author":"DeVore","year":"1989","journal-title":"Manuscripta Math."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB13","first-page":"261","article-title":"Approximation by feed-forward neural networks","volume":"4","author":"DeVore","year":"1997","journal-title":"Ann. Numer. Math."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB14","unstructured":"Y. Gordon, V. Maiorov, M. Meyer, S. Reisner, On best approximation by Ridge functions in the uniform norm, to appear."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB15","series-title":"Approximation Theory, Spline Functions and Applications","first-page":"163","article-title":"Some aspects of radial basis function approximation","author":"Light","year":"1995"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB16","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1006\/jath.1993.1104","article-title":"Fundamentality of Ridge functions","volume":"75","author":"Lin","year":"1993","journal-title":"J. Approx. Theory"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB17","series-title":"Advanced in Computational Mathematics, New Dehi, India","first-page":"257","article-title":"Approximation of multivariate functions","author":"Lin","year":"1994"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB18","first-page":"645","article-title":"Optimal reconstruction of functions from its projections","volume":"42","author":"Logan","year":"1975","journal-title":"Duke Math."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB19","series-title":"Constructive Approximation, Advanced Problems","author":"Lorenz","year":"1996"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB20","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1006\/jath.1998.3304","article-title":"On best approximation by Ridge functions","volume":"99","author":"Maiorov","year":"1999","journal-title":"J. Approx. Theory"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB21","unstructured":"V. Maiorov, On lower bounds in radial basis approximation, submitted for publication."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB22","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1023\/A:1018993908478","article-title":"On the near optimality of the stochastic approximation of smooth functions by neural networks","volume":"13","author":"Maiorov","year":"2000","journal-title":"Adv. in Comput. Math."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB23","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1006\/jath.1998.3305","article-title":"On the approximation of functional classes equipped with a uniform measure using Ridge functions","volume":"99","author":"Maiorov","year":"1999","journal-title":"J. Approx. Theory"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB24","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/S0925-2312(98)00111-8","article-title":"Lower bounds for approximation by MLP neural networks","volume":"25","author":"Maiorov","year":"1999","journal-title":"Neurocomputing"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB25","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/S0166-218X(98)00015-8","article-title":"The degree of approximation of sets in Euclidean space using sets with bounded Vapnik\u2013Chervonenkis dimension","volume":"86","author":"Maiorov","year":"1998","journal-title":"Discrete Appl. Math."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB26","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/s003659900108","article-title":"On the degree of approximation using manifolds of finite pseudo-dimension","volume":"15","author":"Maiorov","year":"1999","journal-title":"Constr. Approx."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB27","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1006\/jath.1996.0031","article-title":"Random approximation and neural networks","volume":"85","author":"Makovoz","year":"1996","journal-title":"J. Approx. Theory"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB28","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1162\/neco.1996.8.1.164","article-title":"Neural networks for optimal approximation of smooth and analytic functions","volume":"8","author":"Mhaskar","year":"1996","journal-title":"Neural Comput."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB29","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/0196-8858(92)90016-P","article-title":"Approximation by superposition of a sigmoidal function and radial basis functions","volume":"13","author":"Mhaskar","year":"1992","journal-title":"Adv. in Appl. Math."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB30","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1023\/A:1018967708053","article-title":"Approximation properties of zonal function networks using scattered data on the sphere","volume":"11","author":"Mhaskar","year":"1999","journal-title":"Adv. Comput. Math."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB31","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1090\/S0002-9939-1964-0161339-9","article-title":"On the Betti numbers of real varieties","volume":"15","author":"Milnor","year":"1964","journal-title":"Proc. Amer. Math. Soc."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB32","first-page":"265","article-title":"Ridge approximation, Chebyshev\u2013Fourier analysis and optimal quadrature formulas","volume":"219","author":"Oskolkov","year":"1997","journal-title":"Proc. Steklov Inst. Math."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB33","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1137\/S0036141097322959","article-title":"Approximation by ridge functions and neural networks","volume":"30","author":"Petrushev","year":"1999","journal-title":"SIAM J. Math. Anal."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB34","unstructured":"A. Pinkus, Open problems in Approximation Theory, International Conference, Voneshta voda, Bulgaria, June 18\u201324, 1993."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB35","first-page":"277","article-title":"Some density problems in multivariate approximation, approximation theory","volume":"86","author":"Pinkus","year":"1995","journal-title":"Proc. IdoMAT Math. Res."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB36","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1006\/jath.1996.0042","article-title":"TDI-Subspaces of C(Rd) and some density problems from neural networks","volume":"85","author":"Pinkus","year":"1996","journal-title":"J. Approx. Theory"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB37","unstructured":"A. Pinkus, Approximation by Ridge Functions, Some Density Problems from Neural Networks, Surface Fitting and Multiresolution Method, Vanderbilt University Press, vol. 2, 1997, pp. 279\u2013292."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB38","doi-asserted-by":"crossref","unstructured":"A. Pinkus, Approximation theory of the MLP model in neural networks, Acta Numer. (1999) 1\u201352.","DOI":"10.1017\/S0962492900002919"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB39","first-page":"193","article-title":"Fractional calculus and wavelet transforms in integral geometry","volume":"1","author":"Rubin","year":"1998","journal-title":"Fractional Calculus Appl. Anal."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB40","unstructured":"B. Rubin, D. Ryabogin, The k-dimensional Radon transform on the n-sphere and related wavelet transforms, preprint."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB41","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/BF02432002","article-title":"Error estimates and conditions numbers for radial basis functions interpolations","volume":"3","author":"Schaback","year":"1995","journal-title":"Adv. in Comput. Math."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB42","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/BF02433047","article-title":"Approximation by radial basis functions with finitely many centers","volume":"12","author":"Schaback","year":"1996","journal-title":"Constr. Approx."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB43","doi-asserted-by":"crossref","unstructured":"E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton, NJ, 1971.","DOI":"10.1515\/9781400883899"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB44","series-title":"Classical Orthogonal Polynomials","author":"Suetin","year":"1979"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB45","unstructured":"G. Szeg\u00f6, Orthogonal Polynomials, 4th Edition, Vol. 23, Amer. Math. Soc. Colloq. Publ., Providence, RI, 1975."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB46","unstructured":"V. Temlyakov, On approximation by ridge functions, Department of Mathematics, University of South Carolina, 1996, 12pp."},{"key":"10.1016\/S0021-9045(02)00011-4_BIB47","series-title":"Theory of Approximation of Function of the Real Variable","author":"Timan","year":"1963"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB48","series-title":"Special Functions and a Theory of Representation of Group","author":"Vilenkin","year":"1965"},{"key":"10.1016\/S0021-9045(02)00011-4_BIB49","series-title":"Trigonometric Series","author":"Zygmund","year":"1959"}],"container-title":["Journal of Approximation Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0021904502000114?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0021904502000114?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,9,18]],"date-time":"2025-09-18T00:14:44Z","timestamp":1758154484000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0021904502000114"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":49,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["S0021904502000114"],"URL":"https:\/\/doi.org\/10.1016\/s0021-9045(02)00011-4","relation":{},"ISSN":["0021-9045"],"issn-type":[{"value":"0021-9045","type":"print"}],"subject":[],"published":{"date-parts":[[2003,1]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"On best approximation of classes by radial functions","name":"articletitle","label":"Article Title"},{"value":"Journal of Approximation Theory","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/S0021-9045(02)00011-4","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"converted-article","name":"content_type","label":"Content Type"},{"value":"Copyright \u00a9 2002 Published by Elsevier Inc.","name":"copyright","label":"Copyright"}]}}