Some numerical reslts on best uniform polynomial approximation of. chi. sup. alpha. on (0,1)

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Let {alpha} be a positive number, and let E{sub n}(chi{sup {alpha}}; (0,1)) denote the error of best uniform approximation to {chi}{sup {alpha}}, by polynomials of degree at most n, on the interval (0,1). The Russian mathematician S.N. Bernstein established the existence of a nonnegative constant {Beta}({alpha}) such that {Beta}({alpha}):= {sub n{yields}{infinity}lim(2n){sup 2{alpha}}E{sub n}({chi}{sup {alpha}};(0.1)). In addition, Bernstein showed that {Beta}{alpha} < {Gamma}(2{alpha}){vert bar}sin(pi}{alpha}){vert bar}/{pi} ({alpha} > 0) and that {Gamma}(2{alpha}){vert bar}sin({pi}{alpha}){vert bar}/{pi} (1{minus}1/2{alpha}{minus}1) < {Beta}({alpha}) ({alpha} > {1/2}), so that the asymptotic behavior of {Beta}({alpha}) is known when {alpha}{yields}{infinity}. Still, the problem of trying to determine {Beta}({alpha}) more precisely, for ... continued below

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Carpenter, A.J. & Varga, R.S. January 1, 1992.

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Let {alpha} be a positive number, and let E{sub n}(chi{sup {alpha}}; (0,1)) denote the error of best uniform approximation to {chi}{sup {alpha}}, by polynomials of degree at most n, on the interval (0,1). The Russian mathematician S.N. Bernstein established the existence of a nonnegative constant {Beta}({alpha}) such that {Beta}({alpha}):= {sub n{yields}{infinity}lim(2n){sup 2{alpha}}E{sub n}({chi}{sup {alpha}};(0.1)). In addition, Bernstein showed that {Beta}{alpha} < {Gamma}(2{alpha}){vert bar}sin(pi}{alpha}){vert bar}/{pi} ({alpha} > 0) and that {Gamma}(2{alpha}){vert bar}sin({pi}{alpha}){vert bar}/{pi} (1{minus}1/2{alpha}{minus}1) < {Beta}({alpha}) ({alpha} > {1/2}), so that the asymptotic behavior of {Beta}({alpha}) is known when {alpha}{yields}{infinity}. Still, the problem of trying to determine {Beta}({alpha}) more precisely, for all {alpha} > 0, is intriguing. To this end, we have rigorously determined the numbers for thirteen values of {alpha}, where these numbers were calculated with a precision of at least 200 significant digits. For each of these thirteen values of {alpha}, Richardson's extrapolation was applied to the products to obtain estimates of {Beta}({alpha}) to approximately 40 decimal places. Included are graphs of the points ({alpha},{Beta}({alpha})) for the thirteen values of {alpha} that we considered.

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Pages: (35 p)

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OSTI; NTIS; GPO Dep.

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  • USSR-US Leningrad conference on approximation theory, Leningrad (USSR), 13-26 May 1991

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  • Other: DE92014859
  • Report No.: ANL/CP-75917
  • Report No.: CONF-9105344--1
  • Grant Number: W-31109-ENG-38
  • Office of Scientific & Technical Information Report Number: 5062082
  • Archival Resource Key: ark:/67531/metadc1053816

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  • January 1, 1992

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  • Jan. 22, 2018, 7:23 a.m.

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  • Jan. 30, 2018, 1:05 p.m.

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Carpenter, A.J. & Varga, R.S. Some numerical reslts on best uniform polynomial approximation of. chi. sup. alpha. on (0,1), article, January 1, 1992; Illinois. (digital.library.unt.edu/ark:/67531/metadc1053816/: accessed October 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.