Some numerical reslts on best uniform polynomial approximation of {chi}{sup {alpha}} on [0,1]

Conference · Wed Jul 01 04:00:00 EDT 1992

OSTI ID:10148456

Carpenter, A J; Varga, R S

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.

• Research Organization: Argonne National Lab., IL (United States)
• Sponsoring Organization: USDOE, Washington, DC (United States)
• DOE Contract Number: W-31109-ENG-38
• OSTI ID: 10148456
• Report Number(s): ANL/CP--75917; CONF-9105344--1; ON: DE92014859
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200
ACCURACY
ALGORITHMS
EXTRAPOLATION
MATHEMATICS AND COMPUTERS
NUMERICAL SOLUTION
POLYNOMIALS