번스타인 상수 (영어 : Bernstein constant )는 1914년 번스타인이 자신의 논문에서 언급한 상수이다. 일반적으로 그리스 문자 베타 (
β
{\displaystyle \beta }
0.2801694990....
{\displaystyle 0.2801694990....}
[1] [2]
이진수
0.01000111101110010011000000110011…
십진수
0.280169499…
십육진수
0.47B930338AAD…
연(속)분수
1
3
+
1
1
+
1
1
+
1
3
+
1
9
+
⋱
{\displaystyle {\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{3+{\cfrac {1}{9+\ddots }}}}}}}}}}}
상한계수 번스타인 상수의 상한계수는 다음과 같다.
k
μ
k
{\displaystyle k\quad \mu _{k}}
0
0.25
{\displaystyle 0\;\;0.25}
1
0.1549083241....
{\displaystyle 1\;\;0.1549083241....}
2
0.1448223214....
{\displaystyle 2\;\;0.1448223214....}
3
0.1422928116....
{\displaystyle 3\;\;0.1422928116....}
4
0.1413408222....
{\displaystyle 4\;\;0.1413408222....}
5
0.1408899963....
{\displaystyle 5\;\;0.1408899963....}
μ
k
=
‖
c
o
s
(
π
t
)
{
F
(
t
)
−
(
a
^
0
(
k
)
+
∑
n
=
1
k
(
2
n
−
1
)
a
^
n
(
k
)
t
2
−
(
2
n
−
1
2
)
2
)
}
‖
L
(
∞
)
[
0
,
+
∞
)
{\displaystyle \mu _{k}={\begin{Vmatrix}cos(\pi t){\begin{Bmatrix}F(t)-\left({\hat {a}}_{0}(k)+\sum _{n=1}^{k}{{(2n-1){\hat {a}}_{n}(k)} \over {t^{2}-\left({{2n-1} \over {2}}\right)^{2}}}\right)\end{Bmatrix}}\end{Vmatrix}}_{L_{(\infty )}[0,+\infty )}}
F
(
t
)
=
t
2
{
ψ
(
t
2
+
3
4
)
−
ψ
(
t
2
+
1
4
)
}
(
t
≥
0
)
{\displaystyle F(t)={t \over 2}{\begin{Bmatrix}\psi \left({t \over 2}+{3 \over 4}\right)-\psi \left({t \over 2}+{1 \over 4}\right)\end{Bmatrix}}\qquad (t\geq 0)}
폴리감마 함수
ψ
a
^
{\displaystyle \psi \quad {\hat {a}}}
추정량
,
‖
‖
{\displaystyle \;\;,\;\;{\begin{Vmatrix}\end{Vmatrix}}}
노름
,
[
0
,
1
)
{\displaystyle ,\;\;[0,1)}
구간 하한계수 번스타인 상수의 하한계수는 다음과 같다.
k
l
k
{\displaystyle k\quad l_{k}}
1
0.2719823590....
{\displaystyle 1\;\;0.2719823590....}
2
0.2789309228....
{\displaystyle 2\;\;0.2789309228....}
3
0.2798110004....
{\displaystyle 3\;\;0.2798110004....}
4
0.2800243339....
{\displaystyle 4\;\;0.2800243339....}
5
0.2800977913....
{\displaystyle 5\;\;0.2800977913....}
l
k
:=
s
u
p
{
B
k
(
λ
1
,
λ
2
,
.
.
.
.
,
λ
k
)
:
{
λ
j
}
j
−
1
k
,
(
j
−
1
<
λ
j
<
j
,
j
≥
1
)
}
{\displaystyle l_{k}:=sup\{B_{k}(\lambda _{1},\lambda _{2},....,\lambda _{k}):\{\lambda _{j}\}_{j-1}^{k}\quad ,\quad (j-1<\lambda _{j}<j,j\geq 1)\}}
B
k
(
λ
1
,
λ
2
,
.
.
.
.
,
λ
k
)
:=
∑
i
=
1
k
φ
k
(
λ
i
)
ψ
k
′
(
λ
i
)
(
1
−
(
2
λ
i
λ
i
+
1
2
)
F
(
λ
i
+
1
2
)
)
∑
i
=
1
k
φ
k
(
λ
i
)
ψ
k
′
(
λ
i
)
(
2
π
λ
i
+
t
a
n
(
π
2
(
λ
i
−
i
+
1
)
)
)
{\displaystyle B_{k}(\lambda _{1},\lambda _{2},....,\lambda _{k}):={{\sum _{i=1}^{k}{{\varphi _{k}(\lambda _{i})} \over {\psi _{k}^{'}(\lambda _{i})}}\left(1-\left({{2\lambda _{i}} \over {\lambda _{i}+{1 \over 2}}}\right)F\left(\lambda _{i}+{1 \over 2}\right)\right)} \over {\sum _{i=1}^{k}{{\varphi _{k}(\lambda _{i})} \over {\psi _{k}^{'}(\lambda _{i})}}\left({{2} \over {\pi \lambda _{i}}}+tan\left({\pi \over 2}(\lambda _{i}-i+1)\right)\right)}}}
φ
k
(
λ
i
)
ψ
k
′
(
λ
i
)
=
∏
j
=
1
i
−
1
(
λ
i
2
−
j
2
)
⋅
∏
j
=
i
k
−
1
(
j
2
−
λ
i
2
)
2
λ
i
∏
j
=
1
i
−
1
(
λ
i
2
−
λ
j
2
)
⋅
∏
j
=
i
+
1
k
(
λ
j
2
−
λ
i
2
)
,
(
1
≤
i
≤
k
)
,
s
u
p
{
.
.
.
.
}
{\displaystyle {{\varphi _{k}(\lambda _{i})} \over {\psi _{k}^{'}(\lambda _{i})}}={{\prod _{j=1}^{i-1}(\lambda _{i}^{2}-j^{2})\cdot \prod _{j=i}^{k-1}(j^{2}-\lambda _{i}^{2})} \over {2\lambda _{i}\prod _{j=1}^{i-1}(\lambda _{i}^{2}-\lambda _{j}^{2})\cdot \prod _{j=i+1}^{k}(\lambda _{j}^{2}-\lambda _{i}^{2})}}\qquad ,\quad (1\leq i\leq k)\;\;,\;\;sup\{....\}}
상한 같이 보기
편집
↑ SORIN G, GAL (2016년 2월 22일). “ON THE BERNSTEIN’S CONSTANT IN CONVEX APPROXIMATION” (PDF) . 《Open Problems in Mathematics》 3 : 1-3. ↑ Bernstein, S. N. (1914), “Sur la meilleure approximation de x par des polynomes de degrés donnés”, 《Acta Math.》 37 : 1–57, doi :10.1007/BF02401828 ↑ Varga, Richard S.; Carpenter, Amos J. (1985년 12월). “On the bernstein conjecture in approximation theory” . 《Constructive Approximation》 1 (1): 333-348. doi :10.1007/BF01890040 . ↑ 가 나 다 라 http://mathworld.wolfram.com/BernsteinsConstant.html ↑ Varga, Richard S.; Carpenter, Amos J. (1987), “A conjecture of S. Bernstein in approximation theory”, 《Math. USSR Sbornik》 57 (2): 547–560, doi :10.1070/SM1987v057n02ABEH003086 , MR 0842399 ↑ http://www.mathnet.ru/links/175d10936abf1e059a1c38f893b4fa60/sm1844.pdf , Richard S. Varga and Amos J. Carpenter, On the Bernstein conjecture in approximation theory, Const. Approx. 1 (1985), 333-348. MR 87g:41066. MR 88f:41030. Zbl. 648.41013. (http://www.math.kent.edu/~varga/pub/paper_150.pdf ) 
![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
