{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:12:45Z","timestamp":1776845565750,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"215","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Consider a (nonnegative) 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                    with support in 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 respective orthogonal polynomials, above a specific index\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script l\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u2113\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ell<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , satisfy a three-term recurrence relation with constant coefficients. We show that the corresponding Stieltjes polynomials, above the index\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 script l minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u2113\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2\\ell -1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , have a very simple and useful representation in terms of the orthogonal polynomials. As a result of this, the Gauss-Kronrod quadrature formulae for\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                    have all the desirable properties, namely, the interlacing of nodes, their inclusion in the closed 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                    (under an additional assumption on\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                    ), and the positivity of all weights. Furthermore, the interpolatory quadrature formulae based on the zeros of the Stieltjes polynomials have positive weights, and both of these quadrature formulae have elevated degrees of exactness.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00732-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"1257-1268","source":"Crossref","is-referenced-by-count":8,"title":["Stieltjes polynomials and related quadrature formulae for a class of weight functions"],"prefix":"10.1090","volume":"65","author":[{"given":"Walter","family":"Gautschi","sequence":"first","affiliation":[]},{"given":"Sotirios","family":"Notaris","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","unstructured":"Wm. R. Allaway, \u201cThe identification of a class of orthogonal polynomial sets\u201d, Ph.D. Thesis, Univ. Alberta, 1972."},{"issue":"176","key":"2","doi-asserted-by":"publisher","first-page":"639","DOI":"10.2307\/2008178","article-title":"On computing Gauss-Kronrod quadrature formulae","volume":"47","author":"Cali\u00f2, Franca","year":"1986","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","series-title":"Mathematics and its Applications, Vol. 13","isbn-type":"print","volume-title":"An introduction to orthogonal polynomials","author":"Chihara, T. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0035-7596","issn-type":"print"},{"issue":"136","key":"9","doi-asserted-by":"publisher","first-page":"812","DOI":"10.2307\/2005400","article-title":"A note on extended Gaussian quadrature rules","volume":"30","author":"Monegato, Giovanni","year":"1976","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1137\/1024039","article-title":"Stieltjes polynomials and related quadrature rules","volume":"24","author":"Monegato, Giovanni","year":"1982","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1016\/0377-0427(90)90355-4","article-title":"Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szeg\u00f6 type. II","volume":"29","author":"Notaris, Sotirios E.","year":"1990","journal-title":"J. Comput. Appl. 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