Modulus of smoothness: Difference between revisions

Revision as of 08:19, 8 April 2015

Revision as of 18:15, 1 September 2015

Line 32:

4. <math>\omega_n(mt)\leq m^n\omega_n(t)</math>, <math>m\in\mathbb{N}</math>, <math>t\geq0.</math>

5. <math>\omega_n(f,\lambda t)\leq (\lambda +1)^n\omega_n(f,\lambda t)</math>, <math>\lambda>0.</math>

6. For <math>r\in N</math>, denote by <math>W^r</math> 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>\|f^{(r)}\|_{L_{\infty}[-1,1]}<+\infty.</math> If <math>f\in W^r</math>, then <math>\omega_r(t,f,[-1,1])\leq t^r\|f^{(r)}\|_{L_{\infty}[-1,1]}, t\geq 0,</math> where <math>\|g(x)\|_{L_{\infty}[-1,1]}={\mathrm{ess} \sup}_{x\in [-1,1]}|g(x)|.</math>