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Best Approximation of Functions like |x|λ exp(−A|x|−α)

✍ Scribed by Michael I Ganzburg

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 392 KB
Volume: 92
Category: Article
ISSN: 0021-9045
DOI: 10.1006/jath.1997.3131

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✦ Synopsis

We determine the exact order of best approximation by polynomials and entire functions of exponential type of functions like . *, : (x)= |x| * exp(&A |x| &: ). In particular, it is shown that E(. *, : , P n , L p (&1, 1))tn &(2*p+:p+2)Â2 p(1+:) _ exp(&(1+: &1 )(A:) 1Â(1+:) cos :?Â2(1+:) n :Â(1+:) ), where E(. *, : , P n , L p (&1, 1)) denotes best polynomial approximation of . *, : in L p (&1, 1), * # R, : # (0, 2], A>0, 1 p . The problem, concerning the exact order of decrease of E(. 0, 2 , P n , L (&1, 1)), has been posed by S. N. Bernstein.

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✦ Synopsis

Best Approximation of Functions like |x|λ exp(−A|x|−α)