{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:39:22Z","timestamp":1776836362405,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"338","license":[{"start":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T00:00:00Z","timestamp":1691020800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We prove that there is no positive measure\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d sigma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">d\\sigma<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    on the interval\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket a comma b right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[a,b]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that the corresponding anti-Gaussian quadrature formula is also a Chebyshev quadrature formula. We also show that the only positive and even measure\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d sigma left-parenthesis t right-parenthesis equals d sigma left-parenthesis negative t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">d\\sigma (t)=d\\sigma (-t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    on the symmetric interval\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket negative a comma a right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[-a,a]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , for which the anti-Gaussian formula has the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"integral Subscript negative a Superscript a Baseline f left-parenthesis t right-parenthesis d sigma left-parenthesis t right-parenthesis equals StartFraction mu 0 Over 2 EndFraction left-bracket f left-parenthesis a right-parenthesis plus f left-parenthesis negative a right-parenthesis right-bracket plus upper R 2 Superscript upper A upper G Baseline left-parenthesis f right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mo>\n                                \u222b\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mi>a<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>a<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mfrac>\n                              <mml:msub>\n                                <mml:mi>\n                                  \u03bc\n                                  \n                                <\/mml:mi>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mn>0<\/mml:mn>\n                                <\/mml:mrow>\n                              <\/mml:msub>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mfrac>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>R<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>A<\/mml:mi>\n                                <mml:mi>G<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\int _{-a}^{a}f(t)d\\sigma (t)=\\frac {\\mu _{0}}{2}[f(a)+f(-a)]+R_{2}^{AG}(f)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n=1<\/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=\"integral Subscript negative a Superscript a Baseline f left-parenthesis t right-parenthesis d sigma left-parenthesis t right-parenthesis equals w 1 f left-parenthesis a right-parenthesis plus w sigma-summation Underscript mu equals 2 Overscript n Endscripts f left-parenthesis t Subscript mu Baseline right-parenthesis plus w 1 f left-parenthesis negative a right-parenthesis plus upper R Subscript n plus 1 Superscript upper A upper G Baseline left-parenthesis f right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mo>\n                                \u222b\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mi>a<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>a<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>w<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>w<\/mml:mi>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bc\n                                  \n                                <\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munderover>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bc\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>w<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>R<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>A<\/mml:mi>\n                                <mml:mi>G<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\int _{-a}^{a}f(t)d\\sigma (t)=w_{1}f(a)+w\\sum _{\\mu =2}^{n}f(t_{\\mu })+w_{1}f(-a)+R_{n+1}^{AG}(f)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for all\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than-or-equal-to 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\geq 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , is the measure\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d sigma left-parenthesis t right-parenthesis equals left-parenthesis a squared minus t squared right-parenthesis Superscript negative 1 slash 2 Baseline d t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">d\\sigma (t)=(a^{2}-t^{2})^{-1\/2}dt<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . It turns out that the formula for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than-or-equal-to 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\geq 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis n minus 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(n-1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -point Gauss-Lobatto quadrature formula for the measure\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d sigma left-parenthesis t right-parenthesis equals left-parenthesis a squared minus t squared right-parenthesis Superscript negative 1 slash 2 Baseline d t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">d\\sigma (t)=(a^{2}-t^{2})^{-1\/2}dt<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , which is a generalization of what happens in the case of the Chebyshev measure of the first kind. Moreover, we compute the anti-Gaussian formulae for any one of the four Chebyshev measures.\n                  <\/p>","DOI":"10.1090\/mcom\/3762","type":"journal-article","created":{"date-parts":[[2022,6,8]],"date-time":"2022-06-08T09:25:40Z","timestamp":1654680340000},"page":"2803-2816","source":"Crossref","is-referenced-by-count":0,"title":["Anti-Gaussian quadrature formulae of Chebyshev type"],"prefix":"10.1090","volume":"91","author":[{"given":"Sotirios","family":"Notaris","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,8,3]]},"reference":[{"key":"1","series-title":"Mathematical Surveys and Monographs","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/178","volume-title":"Quadrature theory","volume":"178","author":"Brass, Helmut","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821853610"},{"key":"2","series-title":"Computer Science and Applied Mathematics","isbn-type":"print","volume-title":"Methods of numerical integration","author":"Davis, Philip J.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0122063600","edition":"2"},{"key":"3","first-page":"100","article-title":"Advances in Chebyshev quadrature","author":"Gautschi, Walter","year":"1976"},{"key":"4","isbn-type":"print","first-page":"72","article-title":"A survey of Gauss-Christoffel quadrature formulae","author":"Gautschi, Walter","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/3764311622"},{"key":"5","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198506720.001.0001","volume-title":"Orthogonal polynomials: computation and approximation","author":"Gautschi, Walter","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0198506724"},{"key":"6","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1090\/S0002-9904-1944-08114-7","article-title":"On Gauss\u2019 and Tchebycheff\u2019s quadrature formulas","volume":"50","author":"Geronimus, J.","year":"1944","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"},{"key":"7","first-page":"655","article-title":"On Gauss\u2019 and Tchebycheff\u2019s quadrature formulae","volume":"51","author":"Geronimus, J.","year":"1946","journal-title":"C. R. (Doklady) Acad. Sci. URSS (N.S.)"},{"key":"8","volume-title":"Approximate calculation of integrals","author":"Krylov, Vladimir Ivanovich","year":"1962"},{"issue":"214","key":"9","doi-asserted-by":"publisher","first-page":"739","DOI":"10.1090\/S0025-5718-96-00713-2","article-title":"Anti-Gaussian quadrature formulas","volume":"65","author":"Laurie, Dirk P.","year":"1996","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"10","isbn-type":"print","volume-title":"Chebyshev polynomials","author":"Mason, J. 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Numer. 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