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\n As one myth of polynomial interpolation and quadrature, Trefethen [Math. Today (Southend-on-Sea) 47 (2011), pp. 184\u2013188] revealed that the Chebyshev interpolation of\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n x<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n a<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n |x-a|<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n (with\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n a<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ><\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n |a|>1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about\n \n \n \n 95<\/mml:mn>\n 95%<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n range of\n \n \n \n \n [<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n [-1,1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n except for a small neighbourhood near the singular point\n \n \n \n \n x<\/mml:mi>\n =<\/mml:mo>\n a<\/mml:mi>\n .<\/mml:mo>\n <\/mml:mrow>\n x=a.<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n In this paper, we rigorously show that the Jacobi expansion for a more general class of\n \n \n \n \n \u03a6\n \n <\/mml:mi>\n \\Phi<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired\n \n \n \n \n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n \\log n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -factor in the pointwise error estimate for the Legendre expansion recently stated in Babus\u0306ka and Hakula [Comput. Methods Appl. Mech Engrg. 345 (2019), pp. 748\u2013773] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates.\n <\/p>","DOI":"10.1090\/mcom\/3835","type":"journal-article","created":{"date-parts":[[2023,2,15]],"date-time":"2023-02-15T09:14:00Z","timestamp":1676452440000},"page":"1747-1778","source":"Crossref","is-referenced-by-count":8,"title":["Pointwise error estimates and local superconvergence of Jacobi expansions"],"prefix":"10.1090","volume":"92","author":[{"given":"Shuhuang","family":"Xiang","sequence":"first","affiliation":[]},{"given":"Desong","family":"Kong","sequence":"additional","affiliation":[]},{"given":"Guidong","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Li-Lian","family":"Wang","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,3,21]]},"reference":[{"key":"1","unstructured":"M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C., 1964."},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1007\/PL00005387","article-title":"Optimal estimates for lower and upper bounds of approximation errors in the \ud835\udc5d-version of the finite element method in two dimensions","volume":"85","author":"Babu\u0161ka, Ivo","year":"2000","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"5","key":"3","doi-asserted-by":"publisher","first-page":"1512","DOI":"10.1137\/S0036142901356551","article-title":"Direct and inverse approximation theorems for the \ud835\udc5d-version of the finite element method in the framework of weighted Besov spaces. I. Approximability of functions in the weighted Besov spaces","volume":"39","author":"Babu\u0161ka, Ivo","year":"2001","journal-title":"SIAM J. Numer. 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Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"6","volume-title":"A treatise on trigonometric series. Vols. I, II","author":"Bary, N. K.","year":"1964"},{"key":"7","unstructured":"S. Bernstein, On the best approximation of |\ud835\udc65-\ud835\udc50|^{\ud835\udc5d}, Doll. Akad. Nauk SSSR 18 (1938), 374\u2013384."},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"1651","DOI":"10.1137\/140996203","article-title":"Superconvergence of discontinuous Galerkin methods for two-dimensional hyperbolic equations","volume":"53","author":"Cao, Waixiang","year":"2015","journal-title":"SIAM J. Numer. 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Purer Appl. 4 (1978), 5\u201356, 377\u2013416."},{"issue":"1-3","key":"13","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/0377-0427(93)90135-X","article-title":"Inequalities for ultraspherical polynomials and application to quadrature","volume":"49","author":"F\u00f6rster, Klaus-J\u00fcrgen","year":"1993","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"14","series-title":"Cambridge Monographs on Applied and Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511618352","volume-title":"Spectral methods for time-dependent problems","volume":"21","author":"Hesthaven, Jan S.","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521792110"},{"key":"15","unstructured":"E. 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