Modulus of smoothness: Difference between revisions

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===Moduli of smoothness===

Modulus of smoothness of order n of a function f∈C[a,b] is the function <math>\omega_n:[0,\infty)\rightarrow\mathbb{R}</math> defined by

<math>\omega_n(t,f,[a,b])=\sup_{h\in[0,t]}\sup_{x\in[a,b-nh]}|\Delta_h^n(f,x)|,</math>

for <math>t\in[0,\frac{b-a}{n}],</math> and

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

<ref>DeVore, Ronald A., Lorentz, George G., Constructive approximation, Springer-Verlag, 1993.</ref>

of a function <math>f\in C[a,b]</math> is the function <math>\omega_n:[0,\infty)\to\mathbb{R}</math> defined by

:<math>\omega_n(t,f,[a,b])=\sup_{h\in[0,t]}\sup_{x\in[a,b-nh]}|\Delta_h^n(f,x)|</math> for <math>t\in[0,\frac{b-a}{n}],</math>

Here we used the definition of the [[https://en.wikipedia.org/wiki/Finite_difference| finite difference]] (n-th order forward difference)

and

<math>\Delta_h^n(f,x_0)=\sum_{i=1}^n(-1)^{n-i}\binom{n}{i} f(x_0+ih).</math>

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

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

:<math>\Delta_h^n(f,x_0)=\sum_{i=1}^n(-1)^{n-i}\binom{n}{i} f(x_0+ih).</math>

===Properties===

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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>

Here <math>\|g(x)\|_{L_{\infty}[-1,1]}=\sup_{x\in [-1,1]}|g(x)|.</math>

===Application===

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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 ''<math>n''</math>, if ''<math>f''</math> is <math>2\pi-</math>-periodic continuous function, there exists a [[trigonometric polynomial]] <math>T_n</math> of degree <math>\le n</math> such that

:<math>|f(x)-T_n(x)|\leq c(k)\omega_k\left(\frac{1}{n},f\right),\quad x\in[0,2\pi],</math>

For every natural number ''n'', if ''f'' is <math>2\pi-</math>periodic continuous function, there exists a [[trigonometric polynomial]] <math>T_n</math> such that

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

<math>|f(x)-T_n(x)|\leq c(k)\omega_k\left(\frac{1}{n},f\right),\quad x\in[0,2\pi],</math>

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

== Mathematical analysis / Moduli of smoothness ==