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==Moduli of smoothness==
The modulus of smoothness of order <math>n</math> <ref>{{cite book |last=DeVore, |first=Ronald A., |last2=Lorentz, |first2=George G., |title=Constructive approximation, |publisher=Springer-Verlag, |series=Grundlehren der mathematischen Wissenschaften |volume=303 |date=1993 |isbn=978-3-642-08075-3 |url=https://link.springer.com/book/9783540506270 |oclc=231539342}}</ref>
of a function <math>f\in C[a,b]</math> is the function <math>\omega_n:[0,\infty)\to\R</math> defined by
where the [[finite difference]] (''n''-th order forward difference) is defined as
:<math>\Delta_h^n(f,x_0)=\sum_{i=10}^n(-1)^{n-i}\binom{n}{i} f(x_0+ih).</math>
==Properties==
*1. <math>\omega_n(0)=0, \omega_n(0+)=0.</math>
*2. <math>\omega_n</math> is non-decreasing on <math>[0,\infty).</math>
*3. <math>\omega_n</math> is continuous on <math>[0,\infty).</math>
*4. For <math>m\in\N, t\geq 0</math> we have:
::<math>\omega_n(mt)\leq m^n\omega_n(t).</math>
*5. <math>\omega_n(f,\lambda t)\leq (\lambda +1)^n\omega_n(f,t),</math> for <math>\lambda>0.</math>
*6. For <math>r\in \N</math> let <math>W^r</math> denote the space of continuous function on <math>[-1,1]</math> that have <math>(r-1)</math>-st absolutely continuous derivative on <math>[-1,1]</math> and
::<math>\left \|f^{(r)} \right \|_{L_{\infty}[-1,1]}<+\infty.</math>
:If <math>f\in W^r,</math> then
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