ベルンシュタイン多項式の微分について
On the derivatives of the Bernstein polynomials
中 村 宗 敬
Munetaka NAKAMURA
Abstract
A representation formula for Bernstein polynomials by Stirling numbers of the first and the second kind are considered to obtain some asymptotic expansions of their derivatives. 1. Introduction
The Bernstein polynomial "N
where （K＝0,1,...,N）is the binomial coefficients as usual; . It is well known that
"N
approximation theorem of Weierstrass. Furthermore Voronovskaya's theorem gives its rate of convergence, that is,
if F is twice continuously differentiable. See [3]. Recently Floater [2] showed
if F is K times continuously differentiable （The convergence is uniform in X). Thus, roughly speaking, the
derivative （ " NF（K） approximates F（K）（X） up to the order .
Here in this paper we ask what becomes of higher rate of convergence, namely, our concern is how behaves for large N following extended form; For analytic F ,
The following representation formula for Bernstein polynomials due to Abel [1] is quite fundamental for the proof;
for R（X）＝XR, where and in general denote the Stirling number of the first and the second kind
respectively, i.e., is the Stirling number of the first kind indicating the number of permutations of degree
M
kind indicating the numbers of ways of partitioning M distinct objects into exactly I blocks ([1]).
2．Stirling numbers
In this section we give explicit representation of both kind through the descent product （X）S＝X（X−1） （X−2） … （X−S＋1） for later use. Firstly we note that it easily follows from the identity X（X＋1）（X＋2） …
for all N and 1 K N ([4]).
Therefore this leads to
where we used a combinatorial identities
These are of course obtained by the original definition, i.e. , by counting the decomposition into the favorable cyclic permutations.
In a similar way the original definition of or a generating function of implies
and so on. In fact, to partition N
and . The number of the first pattern is , while the number of the latter pattern is
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, so that is given by the above equality.
More formally, ([4]) so that we have
3．Main results
Let us precisely state our result.
THEOREM. Suppose that F is analytic in [0,1] . Then
The convergence is uniform in X
Proof. By the assumption one can see that it suffices to prove the equality for each monomial FR（X） ＝XR（R ＝0,1,2,...） . Let us review Abel,s formula in the first section. For each fixed R and sufficiently large N
formula becomes
Recalling the results obtained in the previous section, we can write down the first step sum （in S） for J＝1 as follows.
In case J＝2 we have
ベルンシュタイン多項式の微分について （中村）
Therefore it follows that
Differentiating both sides K times, we obtain
Since the last sum clearly converges to 0 as N References
[1] U.Abel, The complete asymptotic expansion for the Meyer-Konig and Zeller operators, J. Mathematical Analysis and applications 208,109-119, 1997.
[2] V.Kac and P.Cheung, Quantum calculus, Springer, 2002.
[3] H.Oruc and G.M. Philips, A generalization of the Bernstein polynomials, Proc. Edin. Math. Soc. 42, 403-412,1999. [4] G.M. Philips, Interpolation and approximation by polynomials, Canad. Math. Soc. 2003.
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