{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:40:13Z","timestamp":1776793213506,"version":"3.51.2"},"reference-count":28,"publisher":"American Mathematical Society (AMS)","issue":"274","license":[{"start":{"date-parts":[[2011,9,27]],"date-time":"2011-09-27T00:00:00Z","timestamp":1317081600000},"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 In order to reduce the Gibbs phenomenon exhibited by the partial Fourier sums of a periodic function\n \n \n \n f<\/mml:mi>\n f<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , defined on\n \n \n \n \n [<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \u03c0\n \n <\/mml:mi>\n ,<\/mml:mo>\n \n \u03c0\n \n <\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n [-\\pi ,\\pi ]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , discontinuous at 0,\n <\/p>\n

\n Driscoll and Fornberg considered so-called singular Fourier-Pad\u00e9 approximants constructed from the Hermite-Pad\u00e9 approximants of the system of functions\n \n \n \n \n (<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \n g<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n ,<\/mml:mo>\n \n g<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n (1,g_{1} (z),g_{2} (z))<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n \n g<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n g_{1} (z)=\\log (1-z)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n g<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n g_{2} (z)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is analytic, such that\n \n \n \n \n Re<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n \n g<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n \n e<\/mml:mi>\n \n i<\/mml:mi>\n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n )<\/mml:mo>\n =<\/mml:mo>\n f<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\operatorname {Re}(g_{2} (e^{it}))=f (t)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Convincing numerical experiments have been obtained by these authors, but no error estimates have been proven so far. In the present paper we study the special case of Nikishin systems and their Hermite-Pad\u00e9 approximants, both theoretically and numerically. We obtain rates of convergence by using orthogonality properties of the functions involved along with results from logarithmic potential theory. In particular, we address the question of how to choose the degrees of the approximants, by considering diagonal and row sequences, as well as linear Hermite-Pad\u00e9 approximants. Our theoretical findings and numerical experiments confirm that these Hermite-Pad\u00e9 approximants are more efficient than the more elementary Pad\u00e9 approximants, particularly around the discontinuity of the goal function\n \n \n \n f<\/mml:mi>\n f<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-2010-02411-1","type":"journal-article","created":{"date-parts":[[2010,12,29]],"date-time":"2010-12-29T14:39:26Z","timestamp":1293633566000},"page":"931-958","source":"Crossref","is-referenced-by-count":16,"title":["How well does the Hermite\u2013Pad\u00e9 approximation smooth the Gibbs phenomenon?"],"prefix":"10.1090","volume":"80","author":[{"given":"Bernhard","family":"Beckermann","sequence":"first","affiliation":[]},{"given":"Valeriy","family":"Kalyagin","sequence":"additional","affiliation":[]},{"given":"Ana","family":"Matos","sequence":"additional","affiliation":[]},{"given":"Franck","family":"Wielonsky","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2010,9,27]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1070\/SM1999v190n05ABEH000401","article-title":"Strong asymptotics of polynomials of simultaneous orthogonality for Nikishin systems","volume":"190","author":"Aptekarev, A. I.","year":"1999","journal-title":"Mat. Sb.","ISSN":"https:\/\/id.crossref.org\/issn\/0368-8666","issn-type":"print"},{"key":"2","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","volume-title":"Pad\\'{e} approximants. Part I","volume":"13","author":"Baker, George A., Jr.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0201135124"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1016\/j.cam.2007.11.011","article-title":"Reduction of the Gibbs phenomenon for smooth functions with jumps by the \ud835\udf00-algorithm","volume":"219","author":"Beckermann, Bernhard","year":"2008","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"4","unstructured":"J.P. Boyd, Defeating Gibbs phenomenon in Fourier and Chebyshev spectral methods for solving differential equations, in Gibbs Phenomenon, A. 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