{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:38:20Z","timestamp":1776785900678,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"256","license":[{"start":{"date-parts":[[2007,5,23]],"date-time":"2007-05-23T00:00:00Z","timestamp":1179878400000},"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                    A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function. The key to calculating a Gaussian quadrature formula for such a functional is to find the three-term recursion coefficients for the polynomials orthogonal with respect to that functional. We show how to obtain the recursion coefficients by using only the mask coefficients, and without the aid of modified moments. Our result implies the existence of the corresponding refinable functional whenever the mask coefficients are nonnegative, even when the same mask does not define a refinable function. The algorithm requires\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis n squared right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(n^2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    rational operations and, thus, can in principle deliver exact results. Numerical evidence suggests that it is also effective in floating-point arithmetic.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01855-2","type":"journal-article","created":{"date-parts":[[2006,8,16]],"date-time":"2006-08-16T10:28:45Z","timestamp":1155724125000},"page":"1891-1903","source":"Crossref","is-referenced-by-count":9,"title":["Orthogonal polynomials for refinable linear functionals"],"prefix":"10.1090","volume":"75","author":[{"given":"Dirk","family":"Laurie","sequence":"first","affiliation":[]},{"given":"Johan","family":"de Villiers","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,5,23]]},"reference":[{"issue":"12","key":"1","doi-asserted-by":"publisher","first-page":"839","DOI":"10.1002\/1521-4001(200112)81:12<839::AID-ZAMM839>3.0.CO;2-F","article-title":"Some remarks on quadrature formulas for refinable functions and wavelets","volume":"81","author":"Barinka, A.","year":"2001","journal-title":"ZAMM Z. Angew. Math. Mech.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-2267","issn-type":"print"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1023\/A:1019959727304","article-title":"Quadrature formulas for refinable functions and wavelets. II. Error analysis","volume":"4","author":"Barinka, Arne","year":"2002","journal-title":"J. Comput. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1521-1398","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/S0377-0427(98)00116-2","article-title":"How to choose modified moments?","volume":"98","author":"Beckermann, Bernhard","year":"1998","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"4","series-title":"Mathematics and its Applications, Vol. 13","isbn-type":"print","volume-title":"An introduction to orthogonal polynomials","author":"Chihara, T. S.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0677041500"},{"key":"5","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-6333-3","volume-title":"A practical guide to splines","volume":"27","author":"de Boor, Carl","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0387903569"},{"key":"6","isbn-type":"print","first-page":"140","article-title":"Questions of numerical condition related to polynomials","author":"Gautschi, Walter","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0883851261"},{"key":"7","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"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1006\/acha.1999.0306","article-title":"Gauss quadrature for refinable weight functions","volume":"8","author":"Gautschi, Walter","year":"2000","journal-title":"Appl. Comput. Harmon. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-5203","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1006\/acha.1999.0306","article-title":"Gauss quadrature for refinable weight functions","volume":"8","author":"Gautschi, Walter","year":"2000","journal-title":"Appl. Comput. Harmon. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-5203","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/j.cam.2004.10.005","article-title":"Composite quadrature formulae for the approximation of wavelet coefficients of piecewise smooth and singular functions","volume":"180","author":"Huybrechs, Daan","year":"2005","journal-title":"J. Comput. Appl. 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An interactive programming environment for doing formal computations on recursive types, including rational and multiprecision floating-point numbers, polynomials and truncated power series."},{"issue":"126","key":"14","doi-asserted-by":"publisher","first-page":"A1--C4","DOI":"10.2307\/2005948","article-title":"Gauss quadrature rules with \ud835\udc35-spline weight functions","volume":"28","author":"Phillips, James L.","year":"1974","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"15","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1007\/BF01406683","article-title":"An algorithm for Gaussian quadrature given modified moments","volume":"18","author":"Sack, R. A.","year":"1971","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"16","volume-title":"Gaussian quadrature formulas","author":"Stroud, A. 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