Modulus of smoothness: Difference between revisions

Revision as of 19:24, 23 March 2015

Revision as of 18:10, 26 March 2015

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{{one source|date=March 2015}}

{{technical|date=March 2015}}

In [[mathematics]], '''moduli of smoothness''' are used to quantitatively measure smoothness of functions. Moduli of smoothness generalise [[modulus of continuity]] and are used in [[approximation theory]] and [[numerical analysis]] to estimate errors of approximation by [[polynomialspolynomial]]s and [[splinesspline]]s.

==Moduli of smoothness==

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and

:<math>\omega_n(t,f,[a,b])=\omega_n\left(\frac{b-a}{n},f,[a,b]\right),</math>\text{ for <math>}t>\frac{b-a}{n},</math>

where we the [[finite difference]] (n-th order forward difference) are defined as