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Here <math>\|g(x)\|_{L_{\infty}[-1,1]}=\sup_{x\in [-1,1]}|g(x)|.</math>
===Application===
Moduli of smoothness can be used to prove estimates on the error of approximation. Due to property (6), moduli of smoothness provide more general estimates than the estimates in terms of derivatives.
For example, moduli of smoothness are used in Whitney inequality to estimate the error of local polynomial approximation. Another application is given by the more general version of [[https://en.wikipedia.org/wiki/Jackson's_inequality| Jackson inequality
For every natural number n, there exists a constant W(k) such that for any continuous function f on <math>[a,b]</math> we have
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