{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:46:11Z","timestamp":1776725171920,"version":"3.51.2"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"233","license":[{"start":{"date-parts":[[2001,6,12]],"date-time":"2001-06-12T00:00:00Z","timestamp":992304000000},"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                    For the construction of an interpolatory integration rule on the unit circle\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T\">\n                        <mml:semantics>\n                          <mml:mi>T<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">T<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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                    nodes by means of the Laurent polynomials as basis functions for the approximation, we have at our disposal two nonnegative integers\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">p_n<\/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=\"q Subscript n Baseline comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>q<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">q_n,<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p Subscript n Baseline plus q Subscript n Baseline equals n minus 1 comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>q<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>=<\/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>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p_n+q_n=n-1,<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    which determine the subspace of basis functions. The quadrature rule will integrate correctly any function from this subspace. In this paper upper bounds for the remainder term of interpolatory integration rules on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T\">\n                        <mml:semantics>\n                          <mml:mi>T<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">T<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are obtained. These bounds apply to analytic functions up to a finite number of isolated poles outside\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">T.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    In addition, if the integrand function has no poles in the closed unit disc or is a rational function with poles outside\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T\">\n                        <mml:semantics>\n                          <mml:mi>T<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">T<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , we propose a simple rule to determine the value of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">p_n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and hence\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">q_n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in order to minimize the quadrature error term. Several numerical examples are given to illustrate the theoretical results.\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01260-6","type":"journal-article","created":{"date-parts":[[2005,7,11]],"date-time":"2005-07-11T17:02:26Z","timestamp":1121101346000},"page":"281-296","source":"Crossref","is-referenced-by-count":5,"title":["Error bounds for interpolatory quadrature rules on the unit circle"],"prefix":"10.1090","volume":"70","author":[{"given":"J.","family":"Santos-Le\u00f3n","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,6,12]]},"reference":[{"key":"1","unstructured":"A. Bultheel, P. Gonz\u00e1lez-Vera, E. Hendriksen and O. Nj\u00e5stad, Orthogonal rational functions and interpolatory product rules on the unit circle. III: Convergence of general sequences, Preprint."},{"key":"2","unstructured":"A. Bultheel, P. Gonz\u00e1lez-Vera, E. Hendriksen and O. Nj\u00e5stad, Orthogonality and quadrature on the unit circle, IMACS Annals on Computing and Appl. Math. 9 (1991) 205-210."},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1007\/BF02124749","article-title":"Some results about numerical quadrature on the unit circle","volume":"5","author":"Gonz\u00e1lez-Vera, P.","year":"1996","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"4","series-title":"Pure and Applied Mathematics","volume-title":"Applied and computational complex analysis","author":"Henrici, Peter","year":"1974"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/BF01893426","article-title":"Continued fractions associated with trigonometric and other strong moment problems","volume":"2","author":"Jones, William B.","year":"1986","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1112\/blms\/21.2.113","article-title":"Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle","volume":"21","author":"Jones, William B.","year":"1989","journal-title":"Bull. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"key":"7","isbn-type":"print","first-page":"325","article-title":"Bounds for remainder terms in Szeg\u0151 quadrature on the unit circle","author":"Jones, William B.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0817637532"},{"key":"8","doi-asserted-by":"crossref","unstructured":"J.C. Santos-Le\u00f3n, Product rules on the unit circle with uniformly distributed nodes. Error bounds for analytic functions, J. Comput. Appl. Math. 108 (1999) 195-208.","DOI":"10.1016\/S0377-0427(99)00110-7"},{"key":"9","unstructured":"G. Szeg\u00f6, Orthogonal polynomials (Amer. Math. Soc. Providence, R.I., 1939)."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-233\/S0025-5718-00-01260-6\/S0025-5718-00-01260-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-233\/S0025-5718-00-01260-6\/S0025-5718-00-01260-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:30:40Z","timestamp":1776724240000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-233\/S0025-5718-00-01260-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,6,12]]},"references-count":9,"journal-issue":{"issue":"233","published-print":{"date-parts":[[2001,1]]}},"alternative-id":["S0025-5718-00-01260-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-01260-6","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":[[2000,6,12]]}}}