Modulus of smoothness

Moduli of smoothness

The generalization of conception of modulus of continuity is the modulus of smoothness. If for a contionuous function $f$ the modulus of contuniuty is enough small, then the function is constant. Therefore for higher smoothness we can't use such a criterior as a modulus of continuity. Some properties of a function can be described in terms of a modulus of smoothness.

In some problems in the approximation theory the moduli of smoothness are widely used.

Modulus of smoothness of order n of a function $f\in C[a,b]$ is the function $\omega _{n}:[0,\infty )\to \mathbb {R}$ defined by

$\omega _{n}(t,f,[a,b])=\sup _{h\in [0,t]}\sup _{x\in [a,b-nh]}|\Delta _{h}^{n}(f,x)|$ for $t\in [0,{\frac {b-a}{n}}],$

and

$\omega _{n}(t,f,[a,b])=\omega _{n}({\frac {b-a}{n}},f,[a,b]),$ for $t>{\frac {b-a}{n}},$

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

$\Delta _{h}^{n}(f,x_{0})=\sum _{i=1}^{n}(-1)^{n-i}{\binom {n}{i}}f(x_{0}+ih).$

Properties

1. $\omega _{n}(0)=0,$ $\omega _{n}(0+)=0.$

2. $\omega _{n}$ is non-decreasing on $[0,\infty ).$

3. $\omega _{n}$ is continuous on $[0,\infty ).$

4. $\omega _{n}(mt)\leq m^{n}\omega _{n}(t)$, $m\in \mathbb {N}$, $t\geq 0.$

5. $\omega _{n}(f,\lambda t)\leq (\lambda +1)^{n}\omega _{n}(f,\lambda t)$, $\lambda >0.$

6. For $r\in N$, denote by $W^{r}$ the space of continuous function on $[-1,1]$ that have $(r-1)$-st absolutely continuous derivative on $[-1,1]$ and $\|f^{(r)}\|_{L_{\infty }[-1,1]}<+\infty .$ If $f\in W^{r}$, then $\omega _{r}(t,f,[-1,1])\leq t^{r}\|f^{(r)}\|_{L_{\infty }[-1,1]},t\geq 0,$ where $\|g(x)\|_{L_{\infty }[-1,1]}={\mathrm {ess} \sup }_{x\in [-1,1]}|g(x)|.$

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 Jackson inequality:

For every natural number $n$, if $f$ is $2\pi$-periodic continuous function, there exists a trigonometric polynomial $T_{n}$ of degree $\leq n$ such that

$|f(x)-T_{n}(x)|\leq c(k)\omega _{k}\left({\frac {1}{n}},f\right),\quad x\in [0,2\pi ],$

where the constant $c(k)$ depends on $k\in \mathbb {N} .$