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Comp."],"abstract":"<p>\n                    We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis x 1 comma ellipsis comma x Subscript upper N Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mi>N<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(x_1, \\dots , x_N)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x Subscript i Baseline element-of double-struck upper R Superscript d\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>d<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x_i \\in \\mathbb {R}^d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is invariant under permutations of its\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                    arguments. We demonstrate how these symmetries can be exploited to improve the cost versus error ratio in a polynomial approximation of the function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and in particular study the dependence of that ratio on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d comma upper N\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">d, N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and the polynomial degree. These results are then used to construct approximations and prove approximation rates for functions defined on multi-sets where\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                    becomes a parameter of the input.\n                  <\/p>","DOI":"10.1090\/mcom\/3868","type":"journal-article","created":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T09:13:02Z","timestamp":1685524382000},"page":"811-839","source":"Crossref","is-referenced-by-count":7,"title":["Polynomial approximation of symmetric functions"],"prefix":"10.1090","volume":"93","author":[{"given":"Markus","family":"Bachmayr","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Genevi\u00e8ve","family":"Dusson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christoph","family":"Ortner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jack","family":"Thomas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2023,6,28]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1137\/0122012","article-title":"Lower and upper bounds for the number of lattice points in a simplex","volume":"22","author":"Beged-dov, Aharon Gavriel","year":"1972","journal-title":"SIAM J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1399","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"232","DOI":"10.1137\/15M1022628","article-title":"QM\/MM methods for crystalline defects. Part 1: Locality of the tight binding model","volume":"14","author":"Chen, Huajie","year":"2016","journal-title":"Multiscale Model. 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