{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T22:45:45Z","timestamp":1767998745866,"version":"3.49.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,1,10]],"date-time":"2023-01-10T00:00:00Z","timestamp":1673308800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,1,10]],"date-time":"2023-01-10T00:00:00Z","timestamp":1673308800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Justus-Liebig-Universit\u00e4t Gie\u00dfen"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2023,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we compute the spherical Fourier expansion coefficients for the restriction of the generalised Wendland functions from<jats:italic>d<\/jats:italic>-dimensional Euclidean space to the (<jats:italic>d<\/jats:italic>\u2212\u20091)-dimensional unit sphere. We use results from the theory of special functions to show that they can be expressed in a closed form as a multiple of a certain<jats:sub>3<\/jats:sub><jats:italic>F<\/jats:italic><jats:sub>2<\/jats:sub>hypergeometric function. We present tight asymptotic bounds on the decay rate of the spherical Fourier coefficients and, in the case where<jats:italic>d<\/jats:italic>is odd, we are able to provide the precise asymptotic rate of decay. Numerical evidence suggests that this precise asymptotic rate also holds when<jats:italic>d<\/jats:italic>is even and we pose this as an open problem. Finally, we observe a close connection between the asymptotic decay rate of the spherical Fourier coefficients and that of the corresponding Euclidean Fourier transform.<\/jats:p>","DOI":"10.1007\/s10444-022-10005-z","type":"journal-article","created":{"date-parts":[[2023,1,10]],"date-time":"2023-01-10T03:02:57Z","timestamp":1673319777000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Generalised Wendland functions for the sphere"],"prefix":"10.1007","volume":"49","author":[{"given":"Simon","family":"Hubbert","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2209-7240","authenticated-orcid":false,"given":"Janin","family":"J\u00e4ger","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,1,10]]},"reference":[{"key":"10005_CR1","volume-title":"Handbook of Mathematical Functions","author":"M Abramowitz","year":"1964","unstructured":"Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards, Dover (1964)"},{"key":"10005_CR2","doi-asserted-by":"publisher","first-page":"308","DOI":"10.1016\/j.jmva.2018.05.005","volume":"167","author":"A Arafat","year":"2018","unstructured":"Arafat, A., Porcu, E., Bevilacqua, M., Mateu, J.: Equivalence and orthogonality of Gaussian measures on spheres. J. Multivar. Anal. 167, 308\u2013318 (2018)","journal-title":"J. Multivar. Anal."},{"key":"10005_CR3","first-page":"828","volume":"47.2","author":"M Bevilacqua","year":"2019","unstructured":"Bevilacqua, M., Faouzi, T., Furrer, R., Porcu, E.: Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics. Ann. Stat. 47.2, 828\u2013856 (2019)","journal-title":"Ann. Stat."},{"key":"10005_CR4","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1090\/S0025-5718-00-01251-5","volume":"70","author":"MD Buhmann","year":"2001","unstructured":"Buhmann, M.D.: A new class of radial basis functions with compact support. Math. Comput. 70, 307\u2013318 (2001)","journal-title":"Math. Comput."},{"key":"10005_CR5","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/j.jat.2013.09.005","volume":"177","author":"A Chernih","year":"2014","unstructured":"Chernih, A., Hubbert, S.: Closed form representations and properties of the generalised Wendland functions. J. Approx. Theory 177, 17\u201333 (2014)","journal-title":"J. Approx. Theory"},{"key":"10005_CR6","unstructured":"Chernih, A.: Multiscale Wendland radial basis functions and applications to solving partial differential equations. PhD Thesis, University of New South Wales (2013)"},{"key":"10005_CR7","doi-asserted-by":"publisher","first-page":"2733","DOI":"10.1090\/S0002-9939-03-06730-3","volume":"131","author":"D Chen","year":"2003","unstructured":"Chen, D., Menegatto, V.A., Sun, X.: A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc. 131, 2733\u20132740 (2003)","journal-title":"Proc. Amer. Math. Soc."},{"key":"10005_CR8","doi-asserted-by":"crossref","unstructured":"Cuevas, F., Allard, D., Porcu, E.: Fast and exact simulation of Gaussian random fields defined on the sphere cross time. Stat. Comput., 1\u20138 (2019)","DOI":"10.1007\/s11222-019-09873-1"},{"key":"10005_CR9","first-page":"1813","volume":"141","author":"DJ Daley","year":"2013","unstructured":"Daley, D.J., Porcu, E.: Dimension walks and Schoenberg spectral measures. Proc. Amer. Math. Soc. 141, 1813\u20131824 (2013)","journal-title":"Proc. Amer. Math. Soc."},{"key":"10005_CR10","first-page":"399","volume":"12","author":"JC De la Cerda","year":"2018","unstructured":"De la Cerda, J. C., Alegria, A., Porcu, E.: Regularity properties and simulations of Gaussian random fields on the sphere cross time. Electronic Journal of Statistics 12, 399\u2013426 (2018)","journal-title":"Electronic Journal of Statistics"},{"key":"10005_CR11","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1016\/j.spl.2018.07.017","volume":"144","author":"X Emery","year":"2019","unstructured":"Emery, X., Furrer, R., Porcu, E.: A turning bands method for simulating isotropic Gaussian random fields on the sphere. Stat. Probab. Lett. 144, 9\u201315 (2019)","journal-title":"Stat. Probab. Lett."},{"key":"10005_CR12","doi-asserted-by":"crossref","unstructured":"Fasshauer, G.E., McCourt, M.J.: Kernel-based approximation methods using Matlab. Vol. 19 World Scientific Publishing Company (2015)","DOI":"10.1142\/9335"},{"key":"10005_CR13","doi-asserted-by":"publisher","first-page":"593","DOI":"10.1016\/0022-247X(65)90028-4","volume":"12","author":"JL Fields","year":"1965","unstructured":"Fields, J.L.: Asymptotic expansions of a class of hypergeometric polynomials with respect to the order III. J. Math. Anal. Appl. 12, 593\u2013601 (1965)","journal-title":"J. Math. Anal. Appl."},{"issue":"3","key":"10005_CR14","doi-asserted-by":"publisher","first-page":"551","DOI":"10.1137\/0506050","volume":"6","author":"JL Fields","year":"1975","unstructured":"Fields, J.L., Ismail, M.: On the positivity of some 1F2\u2019s,. SIAM J. Math. Anal. 6(3), 551\u2013559 (1975)","journal-title":"SIAM J. Math. Anal."},{"key":"10005_CR15","doi-asserted-by":"publisher","first-page":"1327","DOI":"10.3150\/12-BEJSP06","volume":"19","author":"T Gneiting","year":"2013","unstructured":"Gneiting, T.: Strictly and non-strictly definite functions on spheres. Bernoulli 19, 1327\u20131349 (2013)","journal-title":"Bernoulli"},{"key":"10005_CR16","doi-asserted-by":"crossref","unstructured":"Hubbert, S., Le Gia, Q.T., Morton, T.M.: Spherical radial basis functions, theory and applications, springer briefs in mathematics springer (2015)","DOI":"10.1007\/978-3-319-17939-1"},{"key":"10005_CR17","doi-asserted-by":"publisher","first-page":"999","DOI":"10.1007\/s11004-019-09799-4","volume":"51.8","author":"C Lantu\u00e9joul","year":"2019","unstructured":"Lantu\u00e9joul, C., Freulon, X., Renard, D.: Spectral simulation of isotropic Gaussian random fields on a sphere. Mathematical Geosciences 51.8, 999\u20131020 (2019)","journal-title":"Mathematical Geosciences"},{"key":"10005_CR18","first-page":"3047","volume":"25.6","author":"A Lang","year":"2015","unstructured":"Lang, A., Schwab, C.: Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations. Ann. Appl. Probab. 25.6, 3047\u20133094 (2015)","journal-title":"Ann. Appl. Probab."},{"issue":"Issue 6","key":"10005_CR19","doi-asserted-by":"publisher","first-page":"2065","DOI":"10.1137\/090774550","volume":"48","author":"QT Le Gia","year":"2010","unstructured":"Le Gia, Q.T., Sloan, I.H., Wendland, H.: Multiscale analysis in Sobolev spaces on the sphere. SIAM J. Numer. Anal. 48(Issue 6), 2065\u20132090 (2010)","journal-title":"SIAM J. Numer. Anal."},{"key":"10005_CR20","unstructured":"Luke, Y.L.: Special functions and their approximations: v. 1 Academic Press (1969)"},{"key":"10005_CR21","first-page":"222","volume":"40","author":"AR Miller","year":"1998","unstructured":"Miller, A.R., Srivastava, H.M.: On the Mellin transform of a product of hypergeometric functions. The ANZIAM J. 40, 222\u2013237 (1998)","journal-title":"The ANZIAM J."},{"key":"10005_CR22","doi-asserted-by":"publisher","first-page":"1647","DOI":"10.1090\/S0025-5718-09-02322-9","volume":"79","author":"H Mhaskar","year":"2010","unstructured":"Mhaskar, H., Narcowich, F., Prestin, J., Ward, J.: Lp Bernstein estimates and approximation by spherical basis functions. Math. Comput. 79, 1647\u20131679 (2010)","journal-title":"Math. Comput."},{"key":"10005_CR23","doi-asserted-by":"publisher","first-page":"1393","DOI":"10.1137\/S0036141001395054","volume":"33","author":"FJ Narcowich","year":"2002","unstructured":"Narcowich, F.J., Ward, J.D.: Scattered data interpolation on spheres: error estimates and locally supported basis functions. SIAM J. Math. Anal. 33, 1393\u20131410 (2002)","journal-title":"SIAM J. Math. Anal."},{"key":"10005_CR24","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s10444-005-7506-1","volume":"27","author":"FJ Narcowich","year":"2007","unstructured":"Narcowich, F.J., Sun, X., Ward, J.D.: Approximation power of RBFs and their associated SBFs: a connection. Adv. Comput. Math. 27, 107\u2013124 (2007)","journal-title":"Adv. Comput. Math."},{"key":"10005_CR25","unstructured":"NIST digital library of mathematical functions: http:\/\/dlmf.nist.gov\/, Release 1.1.7 of 2022-10-15 Olver, F.W.J, Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V., Cohl, H.S., McClain, M.A. (eds.) (2022)"},{"key":"10005_CR26","doi-asserted-by":"publisher","first-page":"344","DOI":"10.1111\/insr.12266","volume":"86.2","author":"E Porcu","year":"2018","unstructured":"Porcu, E., Alegria, A., Furrer, R.: Modeling temporally evolving and spatially globally dependent data. Int. Stat. Rev. 86.2, 344\u2013377 (2018)","journal-title":"Int. Stat. Rev."},{"key":"10005_CR27","unstructured":"Prudnikov, A.P., Yu, A.B., Marichev, O.I.: Integrals and Series. Volume 3: More Special Functions, Gordon and Breach Science Publishers (1992)"},{"key":"10005_CR28","doi-asserted-by":"crossref","unstructured":"Larsson, E., Schaback, R.: Scaling of radial basis functions. arXiv:2210.05617 (2022)","DOI":"10.1093\/imanum\/drad035"},{"key":"10005_CR29","doi-asserted-by":"publisher","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":"10005_CR30","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1215\/S0012-7094-42-00908-6","volume":"9","author":"IJ Schoenberg","year":"1942","unstructured":"Schoenberg, I.J.: Positive definite functions on spheres. Duke Math. J. 9, 96\u2013108 (1942)","journal-title":"Duke Math. J."},{"key":"10005_CR31","doi-asserted-by":"crossref","unstructured":"Wendland, H.: Scattered Data Approximation Cambridge University Press (2005)","DOI":"10.1017\/CBO9780511617539"},{"key":"10005_CR32","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1007\/s00211-010-0313-8","volume":"116","author":"H Wendland","year":"2010","unstructured":"Wendland, H.: Multiscale analysis in Sobolev spaces on bounded domains. Numer. Math. 116, 493\u2013517 (2010)","journal-title":"Numer. Math."},{"key":"10005_CR33","doi-asserted-by":"publisher","first-page":"1771","DOI":"10.1070\/SM2002v193n12ABEH000699","volume":"193.12","author":"VP Zastavnyi","year":"2002","unstructured":"Zastavnyi, V.P., Trigub, R.M.: Positive-definite splines of special form. Sbornik: Mathematics 193.12, 1771\u20131800 (2002)","journal-title":"Sbornik: Mathematics"},{"key":"10005_CR34","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1007\/s11253-006-0128-z","volume":"58","author":"VP Zastavnyi","year":"2006","unstructured":"Zastavnyi, V.P.: On some properties of Buhmann functions. Ukr. Math. J. 58, 8 (2006)","journal-title":"Ukr. Math. J."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-022-10005-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-022-10005-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-022-10005-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,4]],"date-time":"2023-12-04T09:31:40Z","timestamp":1701682300000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-022-10005-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,10]]},"references-count":34,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["10005"],"URL":"https:\/\/doi.org\/10.1007\/s10444-022-10005-z","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,10]]},"assertion":[{"value":"23 October 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 December 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 January 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}],"article-number":"3"}}