{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:00:32Z","timestamp":1776787232159,"version":"3.51.2"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"261","license":[{"start":{"date-parts":[[2008,6,20]],"date-time":"2008-06-20T00:00:00Z","timestamp":1213920000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Recently, a fast approximate algorithm for the evaluation of expansions in terms of standard\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper L squared left-parenthesis double-struck upper S squared right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:msup>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"double-struck\">S<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:msup>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathrm {L}^2\\left (\\mathbb {S}^2\\right )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -orthonormal spherical harmonics at arbitrary nodes on the sphere\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper S squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">S<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {S}^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    has been proposed in [S. Kunis and D. Potts. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161:75\u201398, 2003]. The aim of this paper is to develop a new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules. We give a formulation in matrix-vector notation and an explicit factorisation of the spherical Fourier matrix based on the former algorithm. Starting from this, we obtain the corresponding factorisation of the adjoint spherical Fourier matrix and are able to describe the associated algorithm for the adjoint transformation which can be employed to evaluate quadrature rules for arbitrary weights and nodes on the sphere. We provide results of numerical tests showing the stability of the obtained algorithm using as examples classical Gau\u00df-Legendre and Clenshaw-Curtis quadrature rules as well as the\n                    <sans-serif>HEALPix<\/sans-serif>\n                    pixelation scheme and an equidistribution.\n                  <\/p>","DOI":"10.1090\/s0025-5718-07-02029-7","type":"journal-article","created":{"date-parts":[[2007,10,29]],"date-time":"2007-10-29T06:30:33Z","timestamp":1193639433000},"page":"397-419","source":"Crossref","is-referenced-by-count":35,"title":["Fast evaluation of quadrature formulae on the sphere"],"prefix":"10.1090","volume":"77","author":[{"given":"Jens","family":"Keiner","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Potts","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2007,6,20]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1016\/0377-0427(90)90041-W","article-title":"On the associated orthogonal polynomials","volume":"32","author":"Belmehdi, S.","year":"1990","journal-title":"J. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-8858","issn-type":"print"},{"key":"5","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198536826.001.0001","volume-title":"Constructive approximation on the sphere","author":"Freeden, W.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0198536828"},{"key":"6","unstructured":"M. Frigo and S. G. Johnson. FFTW, C subroutine library. http:\/\/www.fftw.org."},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1007\/s00041-003-0018-9","article-title":"FFTs for the 2-sphere-improvements and variations","volume":"9","author":"Healy, D. M., Jr.","year":"2003","journal-title":"J. Fourier Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1069-5869","issn-type":"print"},{"key":"8","unstructured":"J. Keiner, S. Kunis, and D. Potts. NFFT3.0, Softwarepackage, C subroutine library. http:\/\/www.tu-chemnitz.de\/\u223cpotts\/nfft, 2006."},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/S0377-0427(03)00546-6","article-title":"Fast spherical Fourier algorithms","volume":"161","author":"Kunis, Stefan","year":"2003","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"235","key":"10","doi-asserted-by":"publisher","first-page":"1113","DOI":"10.1090\/S0025-5718-00-01240-0","article-title":"Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature","volume":"70","author":"Mhaskar, H. N.","year":"2001","journal-title":"Math. 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