On identities for derivative operators
Keyword(s):
Let $X$ be a commutative algebra with unity $e$ and let $D$ be a derivative on $X$ that means the Leibniz rule is satisfied: $D(f\cdot g)=D(f)\cdot g+f\cdot D(g)$. If $D^{(k)}$ is $k$-th iteration of $D$ then we prove that the following identity holds for any positive integer $k$ $$\frac{1}{k!}\sum\limits_{j=0}^k(-1)^j\binom{k}{j}f^jD^{(m)}(gf^{k-j})=\Phi_{k,m}(g,f)=\begin{cases}0,\ 0\leq m <k,\\ gD(f)^k,\ k=m.\end{cases}$$ As an application we prove a sharp version of Bernstein's inequality for trigonometric polynomials.
Keyword(s):
2001 ◽
Vol 29
(0)
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pp. 125-142
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1996 ◽
pp. 227-238
1995 ◽
Vol 58
(1)
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pp. 15-26
2007 ◽
Vol 271
(3)
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pp. 821-838
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2012 ◽
Vol 37
(2)
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pp. 223-232
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