Modulus of smoothness: Difference between revisions

Revision as of 10:48, 28 September 2018

Revision as of 05:38, 25 September 2020

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==Properties==

*1. <math>\omega_n(0)=0, \omega_n(0+)=0.</math>

*2. <math>\omega_n</math> is non-decreasing on <math>[0,\infty).</math>

*3. <math>\omega_n</math> is continuous on <math>[0,\infty).</math>

*4. For <math>m\in\N, t\geq 0</math> we have:

::<math>\omega_n(mt)\leq m^n\omega_n(t).</math>

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

*6. For <math>r\in \N</math> let <math>W^r</math> denote 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>\left \|f^{(r)} \right \|_{L_{\infty}[-1,1]}<+\infty.</math>

:If <math>f\in W^r,</math> then