Modulus of smoothness

Moduli of smoothness

The generalization of the conception of the modulus of continuity is the modulus of smoothness. If for contionuous function the modulus of contuniuty is enough small, then the function is constant. For higher smoothness we can't use such a criterior as a modulus of continuity. The measure of smoothness of a function can be described in terms of a modulus of smoothness. It is used as a good tool for the elegant estimation of the rate of the approximation function by Bernstein polynomial.

There are some problems in the approximation theory, as the characterization of the best algebraic approximation, for which the moduli of smoothness are commonly 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 $k\in \mathbb {N} .$