{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T13:57:30Z","timestamp":1783000650453,"version":"3.54.5"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"218","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to functions with discontinuities, causing undesirable oscillations over the entire sphere.<\/p>\n                  <p>\n                    Recently, methods for removing the Gibbs phenomenon for one-dimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n                        <mml:semantics>\n                          <mml:mi>N<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    expansion coefficients (either Fourier or Gegenbauer) of a piecewise analytic function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is enough to recover an\n                    <italic>exponentially convergent<\/italic>\n                    approximation to the point values of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in any subinterval in which the function is analytic.\n                  <\/p>\n                  <p>\n                    Here we take a similar approach, proving that knowledge of the first\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n                        <mml:semantics>\n                          <mml:mi>N<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    spherical harmonic coefficients yield an\n                    <italic>exponentially convergent<\/italic>\n                    approximation to a spherical piecewise smooth function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis theta comma phi right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03b8\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03d5\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(\\theta ,\\phi )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in any subinterval\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket theta 1 comma theta 2 right-bracket comma phi element-of left-bracket 0 comma 2 pi right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b8\n                                \n                              <\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b8\n                                \n                              <\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03d5\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[\\theta _1,\\theta _2], \\phi \\in [0,2\\pi ]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where the function is analytic. Thus we entirely overcome the Gibbs phenomenon.\n                  <\/p>","DOI":"10.1090\/s0025-5718-97-00828-4","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"699-717","source":"Crossref","is-referenced-by-count":40,"title":["The resolution of the Gibbs phenomenon for spherical harmonics"],"prefix":"10.1090","volume":"66","author":[{"given":"Anne","family":"Gelb","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[1997]]},"reference":[{"key":"1","unstructured":"M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1970."},{"key":"2","isbn-type":"print","volume-title":"Higher transcendental functions. Vol. I","author":"Erd\u00e9lyi, Arthur","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/089874069X"},{"key":"3","unstructured":"E. Butkov, Mathematical Physics, Addison-Wesley Publishing Company, 1968."},{"key":"4","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970425","volume-title":"Numerical analysis of spectral methods: theory and applications","author":"Gottlieb, David","year":"1977"},{"issue":"1-2","key":"5","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/0377-0427(92)90260-5","article-title":"On the Gibbs phenomenon. I. Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function","volume":"43","author":"Gottlieb, David","year":"1992","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"D. Gottlieb and C.-W. Shu, On The Gibbs Phenomenon III: recovering exponential accuracy in a sub-interval from the spectral partial sum of a piecewise analytic function, SIAM J. Numer. Anal., 33:1 (1996), 280\u2013290.","DOI":"10.1137\/0733015"},{"key":"7","doi-asserted-by":"crossref","unstructured":"D. Gottlieb and C.-W. Shu, On The Gibbs Phenomenon IV: recovering exponential accuracy in a sub-interval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comp. 64:211 (1995), 1081\u20131095.","DOI":"10.1090\/S0025-5718-1995-1284667-0"},{"key":"8","isbn-type":"print","volume-title":"Table of integrals, series, and products","author":"Gradshteyn, I. S.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0122947606"},{"key":"9","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9333-7","volume-title":"Partial differential equations","volume":"1","author":"John, Fritz","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0387906096","edition":"4"},{"key":"10","doi-asserted-by":"crossref","unstructured":"S. Orszag, Fourier Series on Spheres, Mon. Wea. Rev. 102 (1978), 56\u201375.","DOI":"10.1175\/1520-0493(1974)102<0056:FSOS>2.0.CO;2"},{"issue":"6","key":"11","doi-asserted-by":"publisher","first-page":"934","DOI":"10.1137\/0716069","article-title":"On the spectral approximation of discrete scalar and vector functions on the sphere","volume":"16","author":"Swarztrauber, Paul N.","year":"1979","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"12","unstructured":"H. Weyl, Die Gibbssche Erscheinung in der Theorie der Kugelfunktionen, Gesammelte Abhandlungen, Springer-Verlag, 1968, 305\u2013320."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1997-66-218\/S0025-5718-97-00828-4\/S0025-5718-97-00828-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1997-66-218\/S0025-5718-97-00828-4\/S0025-5718-97-00828-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:27:09Z","timestamp":1776720429000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1997-66-218\/S0025-5718-97-00828-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997]]},"references-count":12,"journal-issue":{"issue":"218","published-print":{"date-parts":[[1997,4]]}},"alternative-id":["S0025-5718-97-00828-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-97-00828-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1997]]}}}