{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:27:21Z","timestamp":1776828441191,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"230","license":[{"start":{"date-parts":[[2000,2,24]],"date-time":"2000-02-24T00:00:00Z","timestamp":951350400000},"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":"

\n We consider the convergence of Gauss-type quadrature formulas for the integral\n \n \n \n \n \n \n \u222b\n \n <\/mml:mo>\n 0<\/mml:mn>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:msubsup>\n f<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \u03c9\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n d<\/mml:mi>\n <\/mml:mrow>\n x<\/mml:mi>\n <\/mml:mrow>\n \\int _0^\\infty f(x)\\omega (x)\\mathrm {d}x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n \u03c9\n \n <\/mml:mi>\n \\omega<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a weight function on the half line\n \n \n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n [0,\\infty )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomials\n \n \n \n \n \n \n \u039b\n \n <\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n p<\/mml:mi>\n ,<\/mml:mo>\n q<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n {<\/mml:mo>\n \n \n \u2211\n \n <\/mml:mo>\n \n k<\/mml:mi>\n =<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n \n q<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:munderover>\n \n a<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \n x<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n \\Lambda _{-p,q-1}=\\{\\sum _{k=-p}^{q-1} a_k x^k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n }, where\n \n \n \n \n p<\/mml:mi>\n =<\/mml:mo>\n p<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n p=p(n)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a sequence of integers satisfying\n \n \n \n \n 0<\/mml:mn>\n \n \u2264\n \n <\/mml:mo>\n p<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n \n \u2264\n \n <\/mml:mo>\n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:mrow>\n 0\\le p(n)\\le 2n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n q<\/mml:mi>\n =<\/mml:mo>\n q<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n 2<\/mml:mn>\n n<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n p<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n q=q(n)=2n-p(n)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . It is proved that under certain Carleman-type conditions for the weight and when\n \n \n \n \n p<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n p(n)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or\n \n \n \n \n q<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n q(n)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n goes to\n \n \n \n \n \u221e\n \n <\/mml:mi>\n \\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , then convergence holds for all functions\n \n \n \n f<\/mml:mi>\n f<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for which\n \n \n \n \n f<\/mml:mi>\n \n \u03c9\n \n <\/mml:mi>\n <\/mml:mrow>\n f\\omega<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is integrable on\n \n \n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n [0,\\infty )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss quadrature formulas for the half line.\n <\/p>","DOI":"10.1090\/s0025-5718-99-01107-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:17:31Z","timestamp":1027707451000},"page":"721-747","source":"Crossref","is-referenced-by-count":13,"title":["On the convergence of certain Gauss-type quadrature formulas for unbounded intervals"],"prefix":"10.1090","volume":"69","author":[{"given":"A.","family":"Bultheel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C.","family":"D\u00edaz-Mendoza","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P.","family":"Gonz\u00e1lez-Vera","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R.","family":"Orive","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,2,24]]},"reference":[{"key":"1","volume-title":"Handbook of mathematical functions, with formulas, graphs, and mathematical tables","year":"1966"},{"issue":"1-2","key":"2","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1016\/S0377-0427(96)00122-7","article-title":"Quadrature on the half-line and two-point Pad\u00e9 approximants to Stieltjes functions. 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