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It is well known that using the spherical harmonics up to degree <jats:inline-formula><jats:alternatives><jats:tex-math>$$L \\in {\\mathbb N}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> as ansatz space yields for functions in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {C}^{L+1}(\\mathbb S^{d-1})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>C<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mi>L<\/mml:mi>\n                        <mml:mo>+<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msup>\n                        <mml:mi>S<\/mml:mi>\n                        <mml:mrow>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:mo>-<\/mml:mo>\n                          <mml:mn>1<\/mml:mn>\n                        <\/mml:mrow>\n                      <\/mml:msup>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> the approximation order <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {O}\\left( h^{L+1} \\right) $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mfenced>\n                      <mml:msup>\n                        <mml:mi>h<\/mml:mi>\n                        <mml:mrow>\n                          <mml:mi>L<\/mml:mi>\n                          <mml:mo>+<\/mml:mo>\n                          <mml:mn>1<\/mml:mn>\n                        <\/mml:mrow>\n                      <\/mml:msup>\n                    <\/mml:mfenced>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:italic>h<\/jats:italic> denotes the fill distance of the sampling nodes. In this paper, we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degrees up to <jats:italic>L<\/jats:italic>, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as <jats:inline-formula><jats:alternatives><jats:tex-math>$$h \\rightarrow 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>h<\/mml:mi>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space of the sphere as ansatz space.<\/jats:p>","DOI":"10.1007\/s10444-024-10201-z","type":"journal-article","created":{"date-parts":[[2024,10,22]],"date-time":"2024-10-22T06:02:15Z","timestamp":1729576935000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An optimal ansatz space for moving least squares approximation on spheres"],"prefix":"10.1007","volume":"50","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6342-1799","authenticated-orcid":false,"given":"Ralf","family":"Hielscher","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0375-6535","authenticated-orcid":false,"given":"Tim","family":"P\u00f6schl","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,10,22]]},"reference":[{"key":"10201_CR1","doi-asserted-by":"crossref","unstructured":"Atkinson, K., Han, W.: Spherical harmonics and approximations on the unit sphere: an introduction. 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