{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T05:52:32Z","timestamp":1781502752089,"version":"3.54.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T00:00:00Z","timestamp":1766016000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T00:00:00Z","timestamp":1766016000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["Fondecyt 1251154"],"award-info":[{"award-number":["Fondecyt 1251154"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,5]]},"DOI":"10.1007\/s40314-025-03503-5","type":"journal-article","created":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T07:38:58Z","timestamp":1766043538000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Series expansion and fast simulations of multivariate isotropic random fields on the paraboloid"],"prefix":"10.1007","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9720-7697","authenticated-orcid":false,"given":"Alfredo","family":"Alegr\u00eda","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Galatia","family":"Cleanthous","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Athanasios G.","family":"Georgiadis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,12,18]]},"reference":[{"issue":"1","key":"3503_CR1","doi-asserted-by":"publisher","first-page":"2360","DOI":"10.1214\/21-EJS1842","volume":"15","author":"A Alegr\u00eda","year":"2021","unstructured":"Alegr\u00eda A, Bisiri P, Cleanthous G, Porcu E, White P (2021) Multivariate isotropic random fields on spheres: nonparametric Bayesian modeling and $$L^p$$ fast approximations. Electron J Stat 15(1):2360\u20132392. https:\/\/doi.org\/10.1214\/21-EJS1842","journal-title":"Electron J Stat"},{"issue":"1","key":"3503_CR2","doi-asserted-by":"publisher","first-page":"1394","DOI":"10.1214\/24-EJS2231","volume":"18","author":"A Alegr\u00eda","year":"2024","unstructured":"Alegr\u00eda A, Cleanthous G, Georgiadis AG, Porcu E, White P (2024) Gaussian random fields on the product of spheres: theory and applications. Electron J Stat 18(1):1394\u20131435. https:\/\/doi.org\/10.1214\/24-EJS2231","journal-title":"Electron J Stat"},{"issue":"4","key":"3503_CR3","doi-asserted-by":"publisher","first-page":"2478","DOI":"10.1214\/19-AOS1896","volume":"48","author":"E Anderes","year":"2020","unstructured":"Anderes E, M\u00f8ller J, Rasmussen JG (2020) Isotropic covariance functions on graphs and their edges. Ann Stat 48(4):2478\u20132503. https:\/\/doi.org\/10.1214\/19-AOS1896","journal-title":"Ann Stat"},{"issue":"2","key":"3503_CR4","doi-asserted-by":"publisher","first-page":"1611","DOI":"10.3150\/23-BEJ1647","volume":"30","author":"D Bolin","year":"2024","unstructured":"Bolin D, Simas AB, Wallin J (2024) Gaussian Whittle\u2013Mat\u00e9rn fields on metric graphs. Bernoulli 30(2):1611\u20131639. https:\/\/doi.org\/10.3150\/23-BEJ1647","journal-title":"Bernoulli"},{"key":"3503_CR5","first-page":"12426","volume":"33","author":"V Borovitskiy","year":"2020","unstructured":"Borovitskiy V, Terenin A, Mostowsky P (2020) Mat\u00e9rn Gaussian processes on Riemannian manifolds. Adv Neural Inf Process Syst 33:12426\u201312437","journal-title":"Adv Neural Inf Process Syst"},{"issue":"3","key":"3503_CR6","doi-asserted-by":"publisher","first-page":"1723","DOI":"10.3150\/23-BEJ1650","volume":"30","author":"A Caponera","year":"2024","unstructured":"Caponera A (2024) Asymptotics for isotropic Hilbert-valued spherical random fields. Bernoulli 30(3):1723\u20131745. https:\/\/doi.org\/10.3150\/23-BEJ1650","journal-title":"Bernoulli"},{"issue":"44, 1","key":"3503_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3842\/tsp-2024-28(44)-1","volume":"28","author":"G Cleanthous","year":"2024","unstructured":"Cleanthous G (2024) On the properties of multivariate isotropic random fields on the ball. Theory Stoch Processes 28(44, 1):1\u201325. https:\/\/doi.org\/10.3842\/tsp-2024-28(44)-1","journal-title":"Theory Stoch Processes"},{"issue":"8","key":"3503_CR8","doi-asserted-by":"publisher","first-page":"4873","DOI":"10.1016\/j.spa.2020.02.003","volume":"130","author":"G Cleanthous","year":"2020","unstructured":"Cleanthous G, Georgiadis AG, Lang A, Porcu E (2020) Regularity, continuity and approximation of isotropic Gaussian random fields on compact two-point homogeneous spaces. Stoch Processes Appl 130(8):4873\u20134891. https:\/\/doi.org\/10.1016\/j.spa.2020.02.003","journal-title":"Stoch Processes Appl"},{"issue":"4","key":"3503_CR9","doi-asserted-by":"publisher","first-page":"836","DOI":"10.1007\/s11749-021-00755-1","volume":"30","author":"G Cleanthous","year":"2021","unstructured":"Cleanthous G, Porcu E, White P (2021) Regularity and approximation of time evolving random fields on compact two-point homogeneous spaces. TEST 30(4):836\u2013860. https:\/\/doi.org\/10.1007\/s11749-021-00755-1","journal-title":"TEST"},{"issue":"1","key":"3503_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1515\/mcma-2018-0001","volume":"24","author":"P Creasey","year":"2018","unstructured":"Creasey P, Lang A (2018) Fast generation of isotropic Gaussian random fields on the sphere. Monte Carlo Methods Appl 24(1):1\u201311. https:\/\/doi.org\/10.1515\/mcma-2018-0001","journal-title":"Monte Carlo Methods Appl"},{"key":"3503_CR11","volume-title":"Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications","author":"G Da Prato","year":"1992","unstructured":"Da Prato G, Zabczyk J (1992) Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol 44. Cambridge University Press, Cambridge"},{"key":"3503_CR12","volume-title":"Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics","author":"F Dai","year":"2013","unstructured":"Dai F, Xu Y (2013) Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics. Springer, Berlin"},{"key":"3503_CR13","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1007\/s00211-015-0780-z","volume":"134","author":"PMM De Castro","year":"2016","unstructured":"De Castro PMM, M\u00e9rigot Q, Thibert B (2016) Far-field reflector problem and intersection of paraboloids. Numer Math 134:389\u2013411. https:\/\/doi.org\/10.1007\/s00211-015-0780-z","journal-title":"Numer Math"},{"key":"3503_CR14","first-page":"196","volume":"97","author":"J Du","year":"2025","unstructured":"Du J, Ma C (2025) Time-varying vector random fields on the arcos-quasi-quadratic metric space. Theory Stoch Processes 97:196\u2013225","journal-title":"Theory Stoch Processes"},{"issue":"8","key":"3503_CR15","doi-asserted-by":"publisher","first-page":"1659","DOI":"10.1007\/s00477-019-01717-8","volume":"33","author":"X Emery","year":"2019","unstructured":"Emery X, Porcu E (2019) Simulating isotropic vector-valued Gaussian random fields on the sphere through finite harmonics approximations. Stoch Env Res Risk Assess 33(8):1659\u20131667. https:\/\/doi.org\/10.1007\/s00477-019-01717-8","journal-title":"Stoch Env Res Risk Assess"},{"key":"3503_CR16","doi-asserted-by":"publisher","first-page":"1447","DOI":"10.1007\/s00477-021-02063-4","volume":"36","author":"X Emery","year":"2022","unstructured":"Emery X, Arroyo D, Mery N (2022a) Twenty-two families of multivariate covariance kernels on spheres, with their spectral representations and sufficient validity conditions. Stoch Env Res Risk Assess 36:1447\u20131467. https:\/\/doi.org\/10.1007\/s00477-021-02063-4","journal-title":"Stoch Env Res Risk Assess"},{"issue":"5","key":"3503_CR17","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/s40314-022-01912-4","volume":"41","author":"X Emery","year":"2022","unstructured":"Emery X, Peron AP, Porcu E (2022b) Dimension walks on hyperspheres. Comput Appl Math 41(5):199. https:\/\/doi.org\/10.1007\/s40314-022-01912-4","journal-title":"Comput Appl Math"},{"issue":"1","key":"3503_CR18","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1111\/sapm.12582","volume":"151","author":"M Fasondini","year":"2023","unstructured":"Fasondini M, Olver S, Xu Y (2023) Orthogonal polynomials on a class of planar algebraic curves. Stud Appl Math 151(1):369\u2013405. https:\/\/doi.org\/10.1111\/sapm.12582","journal-title":"Stud Appl Math"},{"key":"3503_CR19","volume-title":"Real Analysis: Modern Techniques and Their Applications","author":"GB Folland","year":"2009","unstructured":"Folland GB (2009) Real Analysis: Modern Techniques and Their Applications. Wiley, New York (MR1681462)"},{"issue":"4","key":"3503_CR20","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1215\/17358787-3649260","volume":"10","author":"JC Guella","year":"2016","unstructured":"Guella JC, Menegatto VA, Peron AP (2016) An extension of a theorem of Schoenberg to products of spheres. Banach J Math Anal 10(4):671\u2013685. https:\/\/doi.org\/10.1215\/17358787-3649260","journal-title":"Banach J Math Anal"},{"key":"3503_CR21","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1016\/j.jmva.2015.08.018","volume":"143","author":"J Guinness","year":"2016","unstructured":"Guinness J, Fuentes M (2016) Isotropic covariance functions on spheres: some properties and modeling considerations. J Multivar Anal 143:143\u2013152. https:\/\/doi.org\/10.1016\/j.jmva.2015.08.018","journal-title":"J Multivar Anal"},{"key":"3503_CR22","first-page":"17160","volume":"34","author":"M Hutchinson","year":"2021","unstructured":"Hutchinson M, Terenin A, Borovitskiy V, Takao S, Teh Y, Deisenroth M (2021) Vector-valued Gaussian processes on Riemannian manifolds via gauge independent projected kernels. Adv Neural Inf Process Syst 34:17160\u201317169","journal-title":"Adv Neural Inf Process Syst"},{"key":"3503_CR23","volume-title":"Probability Theory. A Comprehensive Course","author":"A Klenke","year":"2008","unstructured":"Klenke A (2008) Probability Theory. A Comprehensive Course. Springer Science & Business Media, Berlin"},{"issue":"6","key":"3503_CR24","doi-asserted-by":"publisher","first-page":"3047","DOI":"10.1214\/14-AAP1067","volume":"25","author":"A Lang","year":"2015","unstructured":"Lang A, Schwab C (2015) Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations. Ann Appl Probab 25(6):3047\u20133094. https:\/\/doi.org\/10.1214\/14-AAP1067","journal-title":"Ann Appl Probab"},{"key":"3503_CR25","doi-asserted-by":"publisher","first-page":"185","DOI":"10.31390\/cosa.10.2.04","volume":"10","author":"A Lang","year":"2016","unstructured":"Lang A, Potthoff J, Schlather M, Schwab D (2016) Continuity of random fields on Riemannian manifolds. Commun Stoch Anal 10:185\u2013193. https:\/\/doi.org\/10.31390\/cosa.10.2.04","journal-title":"Commun Stoch Anal"},{"key":"3503_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2019.108583","volume":"156","author":"T Lu","year":"2020","unstructured":"Lu T, Leonenko N, Ma C (2020) Series representations of isotropic vector random fields on balls. Stat Probab Lett 156:108583. https:\/\/doi.org\/10.1016\/j.spl.2019.108583","journal-title":"Stat Probab Lett"},{"key":"3503_CR27","doi-asserted-by":"publisher","first-page":"765","DOI":"10.1007\/s11004-012-9411-8","volume":"44","author":"C Ma","year":"2012","unstructured":"Ma C (2012) Stationary and isotropic vector random fields on spheres. Math Geosci 44:765\u2013778. https:\/\/doi.org\/10.1007\/s11004-012-9411-8","journal-title":"Math Geosci"},{"issue":"3","key":"3503_CR28","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1080\/07362994.2015.1136562","volume":"34","author":"C Ma","year":"2016","unstructured":"Ma C (2016) Stochastic representations of isotropic vector random fields on spheres. Stoch Anal Appl 34(3):389\u2013403. https:\/\/doi.org\/10.1080\/07362994.2015.1136562","journal-title":"Stoch Anal Appl"},{"key":"3503_CR29","doi-asserted-by":"publisher","first-page":"1922","DOI":"10.1007\/s10959-022-01217-6","volume":"36","author":"C Ma","year":"2023","unstructured":"Ma C (2023) Vector random fields on the probability simplex with metric-dependent covariance matrix functions. J Theor Probab 36:1922\u20131938. https:\/\/doi.org\/10.1007\/s10959-022-01217-6","journal-title":"J Theor Probab"},{"issue":"2","key":"3503_CR30","doi-asserted-by":"publisher","first-page":"1299","DOI":"10.1007\/s10959-023-01284-3","volume":"37","author":"C Ma","year":"2024","unstructured":"Ma C (2024) Bifractional Brownian motions on metric spaces. J Theor Probab 37(2):1299\u20131332. https:\/\/doi.org\/10.1007\/s10959-023-01284-3","journal-title":"J Theor Probab"},{"issue":"2","key":"3503_CR31","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1080\/17442508.2024.2425402","volume":"97","author":"C Ma","year":"2025","unstructured":"Ma C (2025) Vector random fields on the arccos-quasi-quadratic metric space. Stochastics 97(2):196\u2013225. https:\/\/doi.org\/10.1080\/17442508.2024.2425402","journal-title":"Stochastics"},{"issue":"1","key":"3503_CR32","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/s10959-018-0872-7","volume":"33","author":"C Ma","year":"2020","unstructured":"Ma C, Malyarenko A (2020) Time-varying isotropic vector random fields on compact two-point homogeneous spaces. J Theor Probab 33(1):319\u2013339. https:\/\/doi.org\/10.1007\/s10959-018-0872-7","journal-title":"J Theor Probab"},{"issue":"4","key":"3503_CR33","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1029\/WR025i004p00605","volume":"25","author":"JR Philip","year":"1989","unstructured":"Philip JR, Knight JH, Waechter RT (1989) The seepage exclusion problem for parabolic and paraboloidal cavities. Water Resour Res 25(4):605\u2013618. https:\/\/doi.org\/10.1029\/WR025i004p00605","journal-title":"Water Resour Res"},{"key":"3503_CR34","volume-title":"Orthogonal Polynomials","author":"G Szeg\u00f6","year":"1967","unstructured":"Szeg\u00f6 G (1967) Orthogonal Polynomials. American Mathematical Society, New York"},{"key":"3503_CR35","doi-asserted-by":"publisher","DOI":"10.1016\/j.cie.2022.108125","volume":"168","author":"EV Vazquez-Carmona","year":"2022","unstructured":"Vazquez-Carmona EV, Vasquez-Gomez JI, Herrera-Lozada JC, Antonio-Cruz M (2022) Coverage path planning for spraying drones. Comput Ind Eng 168:108125. https:\/\/doi.org\/10.1016\/j.cie.2022.108125","journal-title":"Comput Ind Eng"},{"issue":"1","key":"3503_CR36","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1007\/s11854-022-0251-2","volume":"149","author":"Y Xu","year":"2023","unstructured":"Xu Y (2023) Fourier orthogonal series on a paraboloid. J Anal Math\u00e9 149(1):251\u2013279. https:\/\/doi.org\/10.1007\/s11854-022-0251-2","journal-title":"J Anal Math\u00e9"},{"issue":"2","key":"3503_CR37","doi-asserted-by":"publisher","DOI":"10.1111\/sapm.12703","volume":"153","author":"Y Xu","year":"2024","unstructured":"Xu Y (2024) Orthogonal polynomials on domains of revolution. Stud Appl Math 153(2):e12703. https:\/\/doi.org\/10.1111\/sapm.12703","journal-title":"Stud Appl Math"},{"key":"3503_CR38","volume-title":"Spectral Theory of Random Fields","author":"M Yadrenko","year":"1983","unstructured":"Yadrenko M (1983) Spectral Theory of Random Fields. Optimization Software Inc, Publications Division, New York (Translated from the Russian)"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03503-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03503-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03503-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T05:02:27Z","timestamp":1781499747000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03503-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,18]]},"references-count":38,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,5]]}},"alternative-id":["3503"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03503-5","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,18]]},"assertion":[{"value":"9 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 October 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 October 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 December 2025","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 authors do not have any financial or non financial interests that are directly or indirectly related to the work submitted for publication.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"152"}}