{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T03:18:14Z","timestamp":1773890294524,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T00:00:00Z","timestamp":1718064000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T00:00:00Z","timestamp":1718064000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100012740","name":"Gruppo Nazionale per l\u2019Analisi Matematica, la Probabilit\u00e0 e le loro Applicazioni","doi-asserted-by":"publisher","award":["CUP E53C23001670001"],"award-info":[{"award-number":["CUP E53C23001670001"]}],"id":[{"id":"10.13039\/100012740","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100012740","name":"Gruppo Nazionale per l\u2019Analisi Matematica, la Probabilit\u00e0 e le loro Applicazioni","doi-asserted-by":"publisher","award":["CUP E53C23001670001"],"award-info":[{"award-number":["CUP E53C23001670001"]}],"id":[{"id":"10.13039\/100012740","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100012740","name":"Gruppo Nazionale per l\u2019Analisi Matematica, la Probabilit\u00e0 e le loro Applicazioni","doi-asserted-by":"publisher","award":["CUP E53C22001930001"],"award-info":[{"award-number":["CUP E53C22001930001"]}],"id":[{"id":"10.13039\/100012740","id-type":"DOI","asserted-by":"publisher"}]},{"name":"European Union - NextGenerationEU under the Italian Ministry of University and Research","award":["grant ECS00000041 - VITALITY"],"award-info":[{"award-number":["grant ECS00000041 - VITALITY"]}]},{"name":"European Union - NextGenerationEU under the Italian Ministry of University and Research","award":["Project Code: P20229SH29, CUP: J53D23015950001"],"award-info":[{"award-number":["Project Code: P20229SH29, CUP: J53D23015950001"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2024,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, a new family of neural network (NN) operators is introduced. The idea is to consider a Durrmeyer-type version of the widely studied discrete NN operators by Costarelli and Spigler (Neural Netw 44:101\u2013106, 2013). Such operators are constructed using special density functions generated from suitable sigmoidal functions, while the reconstruction coefficients are based on a convolution between a general kernel function <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\chi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c7<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and the function being reconstructed, <jats:italic>f<\/jats:italic>. Here, we investigate their approximation capabilities, establishing both pointwise and uniform convergence theorems for continuous functions. We also provide quantitative estimates for the approximation order thanks to the use of the modulus of continuity of <jats:italic>f<\/jats:italic>; this turns out to be strongly influenced by the asymptotic behaviour of the sigmoidal function <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\sigma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c3<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Our study also shows that the estimates we provide are, under suitable assumptions, the best possible. Finally, <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^p$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mi>p<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-approximation is also established. At the end of the paper, examples of activation functions are discussed.<\/jats:p>","DOI":"10.1007\/s12190-024-02146-9","type":"journal-article","created":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T11:14:12Z","timestamp":1718104452000},"page":"4581-4599","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["The approximation capabilities of Durrmeyer-type neural network operators"],"prefix":"10.1007","volume":"70","author":[{"given":"Lucian","family":"Coroianu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8834-8877","authenticated-orcid":false,"given":"Danilo","family":"Costarelli","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0000-4137-543X","authenticated-orcid":false,"given":"Mariarosaria","family":"Natale","sequence":"additional","affiliation":[]},{"given":"Alexandra","family":"Panti\u015f","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,6,11]]},"reference":[{"key":"2146_CR1","doi-asserted-by":"publisher","first-page":"923","DOI":"10.1002\/mma.1610","volume":"35","author":"G Allasia","year":"2012","unstructured":"Allasia, G., Cavoretto, R., De Rossi, A.: A class of spline functions for landmark-based image registration. Math. Methods Appl. Sci. 35, 923\u2013934 (2012)","journal-title":"Math. Methods Appl. Sci."},{"key":"2146_CR2","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1007\/s40314-013-0011-0","volume":"32","author":"G Allasia","year":"2013","unstructured":"Allasia, G., Cavoretto, R., De Rossi, A.: Lobachevsky spline functions and interpolation to scattered data. Comput. Appl. Math. 32, 71\u201387 (2013)","journal-title":"Comput. Appl. Math."},{"key":"2146_CR3","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1006\/jmaa.1997.5494","volume":"212","author":"GA Anastassiou","year":"1997","unstructured":"Anastassiou, G.A.: Rate of convergence of some neural network operators to the unit-univariate case. J. Math. Anal. Appl. 212, 237\u2013262 (1997)","journal-title":"J. Math. Anal. Appl."},{"key":"2146_CR4","volume-title":"Intelligent Systems: Approximation by Artificial Neural Networks, Intelligent Systems Reference Library","author":"GA Anastassiou","year":"2011","unstructured":"Anastassiou, G.A.: Intelligent Systems: Approximation by Artificial Neural Networks, Intelligent Systems Reference Library, vol. 19. Springer, Berlin (2011)"},{"issue":"14\u201315","key":"2146_CR5","doi-asserted-by":"publisher","first-page":"1702","DOI":"10.1002\/mana.201500225","volume":"289","author":"C Bardaro","year":"2016","unstructured":"Bardaro, C., Faina, L., Mantellini, I.: Quantitative Voronovskaja formulae for generalized Durrmeyer sampling type series. Math. Nachr. 289(14\u201315), 1702\u20131720 (2016)","journal-title":"Math. Nachr."},{"issue":"2","key":"2146_CR6","first-page":"143","volume":"6","author":"C Bardaro","year":"2014","unstructured":"Bardaro, C., Mantellini, I.: Asymptotic expansion of generalized Durrmeyer sampling type series. Jean J. Approx. 6(2), 143\u2013165 (2014)","journal-title":"Jean J. Approx."},{"issue":"15","key":"2146_CR7","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2023.119996","volume":"226","author":"B Baxhaku","year":"2023","unstructured":"Baxhaku, B., Agrawal, P.N.: Neural network operators with hyperbolic tangent functions. Expert Syst. Appl. 226(15), 119996 (2023)","journal-title":"Expert Syst. Appl."},{"issue":"1","key":"2146_CR8","first-page":"101","volume":"14","author":"F Cao","year":"2012","unstructured":"Cao, F., Chen, Z.: The construction and approximation of a class of neural networks operators with ramp functions. J. Comput. Anal. Appl. 14(1), 101\u2013112 (2012)","journal-title":"J. Comput. Anal. Appl."},{"issue":"4","key":"2146_CR9","doi-asserted-by":"publisher","first-page":"626","DOI":"10.1016\/j.neucom.2007.07.024","volume":"71","author":"F Cao","year":"2008","unstructured":"Cao, F., Xie, T., Xu, Z.: The estimate for approximation error of neural networks: a constructive approach. Neurocomputing 71(4), 626\u2013630 (2008)","journal-title":"Neurocomputing"},{"issue":"7","key":"2146_CR10","doi-asserted-by":"publisher","first-page":"1563","DOI":"10.1016\/j.mcm.2008.07.031","volume":"49","author":"F Cao","year":"2009","unstructured":"Cao, F., Zhang, R.: The errors of approximation for feed forward neural networks in the Lp metric. Math. Comput. Model. 49(7), 1563\u20131572 (2009)","journal-title":"Math. Comput. Model."},{"issue":"2","key":"2146_CR11","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1016\/S0893-6080(05)80020-6","volume":"5","author":"P Cardaliaguet","year":"1992","unstructured":"Cardaliaguet, P., Euvrard, G.: Approximation of a function and its derivative with a neural network. Neural Netw. 5(2), 207\u2013220 (1992)","journal-title":"Neural Netw."},{"key":"2146_CR12","doi-asserted-by":"crossref","unstructured":"Chen, H., Yu, D., Li, Z.: The construction and approximation of ReLU neural network operators. J. Funct. Spaces (2022)","DOI":"10.1155\/2022\/1713912"},{"key":"2146_CR13","doi-asserted-by":"crossref","unstructured":"Coroianu, L., Costarelli, D.: Best approximation and inverse results for neural network operators, in print in: Results in Mathematics (2024)","DOI":"10.1007\/s00025-024-02222-3"},{"key":"2146_CR14","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/s00009-022-02138-8","volume":"19","author":"L Coroianu","year":"2022","unstructured":"Coroianu, L., Costarelli, D., Kadak, U.: Quantitative estimates for neural network operators implied by the asymptotic behaviour of the sigmoidal activation functions. Mediterr. J. Math. 19, 211 (2022)","journal-title":"Mediterr. J. Math."},{"key":"2146_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2023.108668","volume":"470","author":"L Coroianu","year":"2023","unstructured":"Coroianu, L., Kadak, U.: Integrating multivariate fuzzy neural networks into fuzzy inference system for enhanced decision making. Fuzzy Sets Syst. 470, 108668 (2023)","journal-title":"Fuzzy Sets Syst."},{"issue":"12","key":"2146_CR16","doi-asserted-by":"publisher","first-page":"1276","DOI":"10.1080\/01630563.2023.2241143","volume":"44","author":"D Costarelli","year":"2023","unstructured":"Costarelli, D., Natale, M., Vinti, G.: Convergence results for nonlinear sampling Kantorovich operators in modular spaces. Numer. Funct. Anal. Optim. 44(12), 1276\u20131299 (2023)","journal-title":"Numer. Funct. Anal. Optim."},{"issue":"1","key":"2146_CR17","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s43670-022-00042-6","volume":"21","author":"D Costarelli","year":"2023","unstructured":"Costarelli, D., Piconi, M., Vinti, G.: Quantitative estimates for Durrmeyer-sampling series in Orlicz spaces. Sampling Th. Signal Proc. Data Anal. 21(1), 3 (2023)","journal-title":"Sampling Th. Signal Proc. Data Anal."},{"key":"2146_CR18","doi-asserted-by":"publisher","first-page":"5069","DOI":"10.1007\/s00521-018-03998-6","volume":"31","author":"D Costarelli","year":"2019","unstructured":"Costarelli, D., Sambucini, A.R., Vinti, G.: Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications. Neural Comput. Appl. 31, 5069\u20135078 (2019)","journal-title":"Neural Comput. Appl."},{"key":"2146_CR19","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.neunet.2013.03.015","volume":"44","author":"D Costarelli","year":"2013","unstructured":"Costarelli, D., Spigler, R.: Approximation results for neural network operators activated by sigmoidal functions. Neural Netw. 44, 101\u2013106 (2013)","journal-title":"Neural Netw."},{"key":"2146_CR20","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1016\/j.jat.2014.06.004","volume":"185","author":"D Costarelli","year":"2014","unstructured":"Costarelli, D., Spigler, R.: Convergence of a family of neural network operators of the Kantorovich type. J. Approx. Theory 185, 80\u201390 (2014)","journal-title":"J. Approx. Theory"},{"key":"2146_CR21","doi-asserted-by":"publisher","first-page":"4323","DOI":"10.1090\/tran\/7155","volume":"370","author":"D Cruz-Uribe","year":"2018","unstructured":"Cruz-Uribe, D., Hasto, P.: Extrapolation and interpolation in generalized Orlicz spaces. Trans. Am. Math. Soc. 370, 4323\u20134349 (2018)","journal-title":"Trans. Am. Math. Soc."},{"key":"2146_CR22","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/BF02551274","volume":"2","author":"G Cybenko","year":"1989","unstructured":"Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2, 303\u2013314 (1989)","journal-title":"Math. Control Signals Syst."},{"issue":"4","key":"2146_CR23","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1016\/0021-9045(81)90101-5","volume":"31","author":"MM Derriennic","year":"1981","unstructured":"Derriennic, M.M.: Sur l\u2019approximation de fonctions integrables sur $$[0,1]$$ par des plynomes de Berstein modifies. J. Approx. Theory 31(4), 325\u2013343 (1981)","journal-title":"J. Approx. Theory"},{"key":"2146_CR24","unstructured":"Durrmeyer, J.L.: Une formule d\u2019inversion de la transform\u00e9e de Laplace: applications \u00e0 la th\u00e9orie des moments, Th\u00e9se de 3e cycle, Universit\u00e9 de Paris (1967)"},{"issue":"1\u20132","key":"2146_CR25","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1016\/j.jat.2008.11.002","volume":"160","author":"H Gonska","year":"2009","unstructured":"Gonska, H., Heilmann, M., Rasa, I.: Convergence of iterates of genuine and ultraspherical Durrmeyer operators to the limiting semigroup: $$c^2$$-estimates. J. Approx. Theory 160(1\u20132), 243\u2013255 (2009)","journal-title":"J. Approx. Theory"},{"issue":"3\u20134","key":"2146_CR26","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/s00025-012-0287-1","volume":"62","author":"H Gonska","year":"2012","unstructured":"Gonska, H., Kacso, D., Rasa, I.: The genuine Bernstein\u2013Durrmeyer operators revisited. Results Math. 62(3\u20134), 295\u2013310 (2012)","journal-title":"Results Math."},{"key":"2146_CR27","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1016\/0021-9045(91)90004-T","volume":"67","author":"H Gonska","year":"1991","unstructured":"Gonska, H., Zhou, X.: A global inverse theorem on simultaneous approximation by Berstein\u2013Durrmeyer operators. J. Approx. Theory 67, 284\u2013302 (1991)","journal-title":"J. Approx. Theory"},{"key":"2146_CR28","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/j.neunet.2020.11.010","volume":"134","author":"I G\u00fchring","year":"2021","unstructured":"G\u00fchring, I., Raslan, M.: Approximation rates for neural networks with encodable weights in smoothness spaces. Neural Netw. 134, 107\u2013130 (2021)","journal-title":"Neural Netw."},{"key":"2146_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00025-018-0932-4","volume":"74","author":"M Heilmann","year":"2019","unstructured":"Heilmann, M., Rasa, I.: A nice representation for a link between Baskakov- and Sz\u00e1sz\u2013Mirakjan\u2013Durrmeyer operators and their Kantorovich variants. Results Math. 74, 1\u201312 (2019)","journal-title":"Results Math."},{"key":"2146_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2022.114426","volume":"414","author":"U Kadak","year":"2022","unstructured":"Kadak, U.: Multivariate neural network interpolation operators. J. Comput. Appl. Math. 414, 114426 (2022)","journal-title":"J. Comput. Appl. Math."},{"key":"2146_CR31","doi-asserted-by":"publisher","DOI":"10.1016\/j.jat.2020.105435","volume":"257","author":"PC Kainen","year":"2020","unstructured":"Kainen, P.C., Kurkov\u00e1, V., Vogt, A.: Approximative compactness of linear combinations of characteristic functions. J. Approx. Theory 257, 105435 (2020)","journal-title":"J. Approx. Theory"},{"issue":"9","key":"2146_CR32","doi-asserted-by":"publisher","first-page":"2746","DOI":"10.1109\/TNNLS.2018.2888517","volume":"30","author":"V Kurkov\u00e1","year":"2019","unstructured":"Kurkov\u00e1, V., Sanguineti, M.: Classification by sparse neural networks. IEEE Trans. Neural Netw. Learn. Syst. 30(9), 2746\u20132754 (2019)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"2146_CR33","volume":"418","author":"Y Qian","year":"2022","unstructured":"Qian, Y., Yu, D.: Rates of approximation by neural network interpolation operators. Appl. Math. Comput. 418, 126781 (2022)","journal-title":"Appl. Math. Comput."},{"issue":"4","key":"2146_CR34","doi-asserted-by":"publisher","first-page":"791","DOI":"10.1142\/S0219530521500123","volume":"20","author":"Y Qian","year":"2022","unstructured":"Qian, Y., Yu, D.: Neural network interpolation operators activated by smooth ramp functions. Anal. Appl. 20(4), 791\u2013813 (2022)","journal-title":"Anal. Appl."},{"key":"2146_CR35","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/j.neunet.2014.09.003","volume":"61","author":"J Schmidhuber","year":"2015","unstructured":"Schmidhuber, J.: Deep learning in neural networks: an overview. Neural Netw. 61, 85\u2013117 (2015)","journal-title":"Neural Netw."},{"issue":"2","key":"2146_CR36","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2023.128009","volume":"533","author":"M Sharma","year":"2024","unstructured":"Sharma, M., Singh, U.: Some density results by deep Kantorovich type neural network operators. J. Math. Anal. Appl. 533(2), 128009 (2024)","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"2146_CR37","doi-asserted-by":"publisher","first-page":"787","DOI":"10.1016\/j.acha.2019.06.004","volume":"48","author":"DX Zhou","year":"2020","unstructured":"Zhou, D.X.: Universality of deep convolutional neural networks. Appl. Comput. Harmonic Anal. 48(2), 787\u2013794 (2020)","journal-title":"Appl. Comput. Harmonic Anal."},{"key":"2146_CR38","volume-title":"Neural Approximations for Optimal Control and Decision, Communications and Control Engineering book series CCE","author":"R Zoppoli","year":"2020","unstructured":"Zoppoli, R., Sanguineti, M., Gnecco, G., Parisini, T.: Neural Approximations for Optimal Control and Decision, Communications and Control Engineering book series CCE. Springer, Cham (2020)"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-024-02146-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-024-02146-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-024-02146-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,26]],"date-time":"2024-09-26T12:28:39Z","timestamp":1727353719000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-024-02146-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,11]]},"references-count":38,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["2146"],"URL":"https:\/\/doi.org\/10.1007\/s12190-024-02146-9","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,11]]},"assertion":[{"value":"6 May 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 May 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 May 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 June 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author declares that he has no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}