Modulus of smoothness: Difference between revisions

Revision as of 07:24, 5 December 2014

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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'', thereif exists''f'' ais constant<math>2\pi-</math>periodic W(k)continuous suchfunction, thatthere forexists anya continuous[[trigonometric function f onpolynomial]] <math>[a,b]T_n</math> wesuch havethat

<math>E_{n|f(x)-1}T_n(fx)_{[a,b]}|\leq Wc(k)\omega_k\left(\frac{b-a1}{kn},f\right),\quad x\in[a0,b2\pi]\right).,</math>

where constant <math>W(k)</math> depends on <math>k\in\mathbb{N}.</math>

== Mathematical analysis / Moduli of smoothness ==