{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:23:31Z","timestamp":1776785011413,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"254","license":[{"start":{"date-parts":[[2007,1,23]],"date-time":"2007-01-23T00:00:00Z","timestamp":1169510400000},"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 discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the function is globally smooth, while the presence of jump discontinuities is responsible for spurious\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper O left-parenthesis 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/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\">{\\mathcal O}(1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    Gibbs\u2019 oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order. Classical filters and mollifiers are constructed to have compact support in the Fourier (frequency) and physical (time) spaces respectively, and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region. Here we construct a noncompactly supported filter and mollifier with optimal\n                    <italic>joint<\/italic>\n                    time-frequency localization for a given number of vanishing moments, resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains. Not giving preference to either space allows for a more balanced error decomposition, which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters, yet obtain true exponential accuracy.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01822-9","type":"journal-article","created":{"date-parts":[[2006,2,15]],"date-time":"2006-02-15T11:05:20Z","timestamp":1140001520000},"page":"767-790","source":"Crossref","is-referenced-by-count":37,"title":["Optimal filter and mollifier for piecewise smooth spectral data"],"prefix":"10.1090","volume":"75","author":[{"given":"Jared","family":"Tanner","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,1,23]]},"reference":[{"issue":"2-3","key":"1","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1016\/0096-3003(94)00180-C","article-title":"A lag-averaged generalization of Euler\u2019s method for accelerating series","volume":"72","author":"Boyd, John P.","year":"1995","journal-title":"Appl. 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Gelb and Zackiewicz, Determining Analyticity for Parameter Optimization of the Gegenbauer Reconstruction Method, preprint."},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1006\/acha.1999.0262","article-title":"Detection of edges in spectral data","volume":"7","author":"Gelb, Anne","year":"1999","journal-title":"Appl. Comput. Harmon. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-5203","issn-type":"print"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"1389","DOI":"10.1137\/S0036142999359153","article-title":"Detection of edges in spectral data. II. Nonlinear enhancement","volume":"38","author":"Gelb, Anne","year":"2000","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"12","unstructured":"A. Gelb and J. Tanner, Robust Reprojection Methods for the Resolution of the Gibbs Phenomenon, ACHA, to appear."},{"issue":"211","key":"13","doi-asserted-by":"publisher","first-page":"1081","DOI":"10.2307\/2153484","article-title":"On the Gibbs phenomenon. IV. Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function","volume":"64","author":"Gottlieb, David","year":"1995","journal-title":"Math. 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