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| | I'm Tanesha (23) from Old Bridge Of Urr, United Kingdom. <br>I'm learning Dutch literature at a local university and I'm just about to graduate.<br>I have a part time job in a university.<br><br>Feel free to visit my homepage; [http://topne.ws/WordpressBackup442229 wordpress dropbox backup] |
| |[[Binary numeral system|Binary]]
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| | 0.01000111101110010011000000110011…
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| |-
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| | [[Decimal]]
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| | 0.280169499…
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| | [[Hexadecimal]]
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| | 0.47B930338AAD…
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| |-
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| | [[Continued fraction]]
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| | <math>\cfrac{1}{3 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{3 + \cfrac{1}{9+ \ddots}}}}}</math>
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| '''Bernstein's constant''', usually denoted by the Greek letter β ([[Beta (letter)|beta]]), is a [[mathematical constant]] named after [[Sergei Natanovich Bernstein]] and is approximately equal to 0.2801694990. | |
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| == Definition ==
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| Let ''E''<sub>''n''</sub>(ƒ) be the error of the best [[uniform approximation]] to a [[real function]] ''ƒ''(''x'') on the interval [−1, 1] by real polynomials of no more than degree ''n''. In the case of ''ƒ''(''x'') = |''x''|, {{harvtxt|Bernstein|1914}} showed that the limit
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| :<math>\beta=\lim_{n \to \infty}2nE_{2n}(f),\,</math>
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| called '''Bernstein's constant''', exists and is between 0.278 and 0.286. His [[conjecture]] that the limit is:
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| :<math>\frac {1}{2\sqrt {\pi}}=0.28209\dots\,.</math> | |
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| was disproven by {{harvtxt|Varga|Carpenter|1987}}, who calculated
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| :<math>\beta=0.280169499023\dots\,.</math>
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| == References ==
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| * {{citation|title=Sur la meilleure approximation de ''x'' par des polynomes de degrés donnés|last=Bernstein|first= S. N. |journal= Acta Math. |volume=37|pages= 1–57|year= 1914 |doi=10.1007/BF02401828}}
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| * {{citation|last=Varga|first= Richard S.|last2= Carpenter|first2= Amos J. |title=A conjecture of S. Bernstein in approximation theory|journal= Math. USSR Sbornik |volume=57|pages= 547–560|year= 1987|mr=0842399|doi = 10.1070/SM1987v057n02ABEH003086|issue=2}}
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| * {{MathWorld |urlname=BernsteinsConstant |title=Bernstein's Constant}}
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| [[Category:Numerical analysis]]
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| [[Category:Mathematical constants]]
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I'm Tanesha (23) from Old Bridge Of Urr, United Kingdom.
I'm learning Dutch literature at a local university and I'm just about to graduate.
I have a part time job in a university.
Feel free to visit my homepage; wordpress dropbox backup