{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:38:07Z","timestamp":1776724687230,"version":"3.51.2"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"224","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    The usual way to determine the asymptotic behavior of the Chebyshev coefficients for a function is to apply the method of steepest descent to the integral representation of the coefficients. However, the procedure is usually laborious. We prove an asymptotic upper bound on the Chebyshev coefficients for the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k Superscript t h\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>h<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">k^{th}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    integral of a function. The tightness of this upper bound is then analyzed for the case\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k=1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the first integral of a function. It is shown that for geometrically converging Chebyshev series the theorem gives the tightest upper bound possible as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n right-arrow normal infinity\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\rightarrow \\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . For functions that are singular at the endpoints of the Chebyshev interval,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x equals plus-or-minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x=\\pm 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the theorem is weakened. Two examples are given. In the first example, we apply the method of steepest descent to directly determine (laboriously!) the asymptotic Chebyshev coefficients for a function whose asymptotics have not been given previously in the literature: a Gaussian with a maximum at an endpoint of the expansion interval. We then easily obtain the asymptotic behavior of its first integral, the error function, through the application of the theorem. The second example shows the theorem is weakened for functions that are regular except at\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x equals plus-or-minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x=\\pm 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We conjecture that it is only for this class of functions that the theorem gives a poor upper bound.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00976-4","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1601-1616","source":"Crossref","is-referenced-by-count":2,"title":["Asymptotic upper bounds for the coefficients in the Chebyshev series expansion for a general order integral of a function"],"prefix":"10.1090","volume":"67","author":[{"given":"Natasha","family":"Flyer","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","isbn-type":"print","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0486612724"},{"key":"2","volume-title":"Mathematical analysis","author":"Apostol, Tom M.","year":"1974","edition":"2"},{"key":"3","series-title":"International Series in Pure and Applied Mathematics","isbn-type":"print","volume-title":"Advanced mathematical methods for scientists and engineers","author":"Bender, Carl M.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/007004452X"},{"key":"4","doi-asserted-by":"crossref","unstructured":"J.P. Boyd, The rate of convergence of Fourier coefficients for entire functions of infinite order with application to the Weideman-Cloot sinh-mapping for pseudospectral computations on an infinite interval, J. Comp. Phys. vol. 110, 1994, pp. 360-375.","DOI":"10.1006\/jcph.1994.1032"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1016\/0021-9991(84)90124-4","article-title":"Asymptotic coefficients of Hermite function series","volume":"54","author":"Boyd, John P.","year":"1984","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"J.P. Boyd, Chebyshev and Fourier Spectral Methods, Springer-Verlag, New York, 1989.","DOI":"10.1007\/978-3-642-83876-7"},{"key":"7","doi-asserted-by":"publisher","first-page":"25","DOI":"10.2307\/2004092","article-title":"Some estimates of the coefficients in the Chebyshev series expansion of a function","volume":"19","author":"Elliott, David","year":"1965","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"274","DOI":"10.2307\/2003301","article-title":"The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function","volume":"18","author":"Elliott, David","year":"1964","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"9","volume-title":"Chebyshev polynomials in numerical analysis","author":"Fox, L.","year":"1968"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1998-67-224\/S0025-5718-98-00976-4\/S0025-5718-98-00976-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-224\/S0025-5718-98-00976-4\/S0025-5718-98-00976-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:51:52Z","timestamp":1776721912000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-224\/S0025-5718-98-00976-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":9,"journal-issue":{"issue":"224","published-print":{"date-parts":[[1998,10]]}},"alternative-id":["S0025-5718-98-00976-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-00976-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1998]]}}}