Modulus of smoothness: Difference between revisions

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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

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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==