bernstein’s inequality
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2020 ◽  
Vol 13 (4) ◽  
pp. 503-514 ◽  
Author(s):  
Carlo Bardaro ◽  
Paul L. Butzer ◽  
Ilaria Mantellini ◽  
Gerhard Schmeisser

Abstract We establish a general version of Cauchy’s integral formula and a residue theorem for polar-analytic functions, employing the new notion of logarithmic poles. As an application, a Boas-type differentiation formula in Mellin setting and a Bernstein-type inequality for polar Mellin derivatives are deduced.


2019 ◽  
Vol 7 (4) ◽  
pp. 13-16
Author(s):  
Mirosław Baran ◽  
Paweł Ozorka

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.


2019 ◽  
Vol 9 (3) ◽  
pp. 1181-1207
Author(s):  
H. Queffélec ◽  
R. Zarouf

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