Modulus of smoothness

Moduli of smoothness

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 $k\in \mathbb {N} .$