{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:36:18Z","timestamp":1776785778685,"version":"3.51.2"},"reference-count":7,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,5,1]],"date-time":"2007-05-01T00:00:00Z","timestamp":1177977600000},"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 evaluate explicitly the integrals\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"integral Subscript negative 1 Superscript 1 Baseline pi Subscript n Baseline left-parenthesis t right-parenthesis slash left-parenthesis r minus-or-plus t right-parenthesis d t comma StartAbsoluteValue r EndAbsoluteValue not-equals 1\">\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:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03c0\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>\n                              \u2213\n                              \n                            <\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mtext>\u00a0<\/mml:mtext>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\n                              \u2260\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\int _{-1}^{1}\\pi _{n}(t)\/(r\\mp t)dt,\\ |r|\\neq 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , with the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"pi Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\pi _{n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    being any one of the four Chebyshev polynomials of degree\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                    . These integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with Chebyshev abscissae, when the function to be integrated is analytic in a domain containing\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket negative 1 comma 1 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:mn>1<\/mml:mn>\n                            <mml:mo>,<\/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\">[-1,1]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in its interior.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01859-x","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1217-1231","source":"Crossref","is-referenced-by-count":15,"title":["Integral formulas for Chebyshev polynomials and the error term of interpolatory quadrature formulae for analytic functions"],"prefix":"10.1090","volume":"75","author":[{"given":"Sotirios","family":"Notaris","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,5,1]]},"reference":[{"key":"1","unstructured":"G. Akrivis, Fehlerabsch\u00e4tzungen bei der numerischen Integration in einer und mehreren Dimensionen, Doctoral Dissertation, Ludwig-Maximilians-Universit\u00e4t M\u00fcnchen, 1982."},{"key":"2","volume-title":"The elements of real analysis","author":"Bartle, Robert G.","year":"1976","edition":"2"},{"key":"3","doi-asserted-by":"publisher","first-page":"863","DOI":"10.2307\/2004620","article-title":"Error estimates for a Chebyshev quadrature method","volume":"24","author":"Basu, N. K.","year":"1970","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","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":"5","isbn-type":"print","first-page":"133","article-title":"Remainder estimates for analytic functions","author":"Gautschi, Walter","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0792315839"},{"key":"6","first-page":"153","article-title":"Fehlerabsch\u00e4tzung bei numerischer Integration nach Gauss","author":"H\u00e4mmerlin, G.","year":"1972"},{"issue":"1-2","key":"7","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1016\/S0377-0427(00)00672-5","article-title":"Interpolatory quadrature formulae with Chebyshev abscissae","volume":"133","author":"Notaris, Sotirios E.","year":"2001","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01859-X\/S0025-5718-06-01859-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01859-X\/S0025-5718-06-01859-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:36:12Z","timestamp":1776782172000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01859-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,5,1]]},"references-count":7,"journal-issue":{"issue":"255","published-print":{"date-parts":[[2006,7]]}},"alternative-id":["S0025-5718-06-01859-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01859-x","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":[[2006,5,1]]}}}