Let \(f\in C_B(I)\). The modulus of continuity \(\omega _f\in \mathbb{R}^{[0,\infty )}\) of the function \(f\) is defined by
$$\omega _f(\delta )=\sup_{x,y\in I\\ |x-y|\le \delta}|f(x)-f(y)| =\sup_{|h|\le \delta \\ x,x+h\in I}|\Delta _hf(x)|,\ \delta \ge 0.
$$
More generally, if \(f:I\to \mathbb{R}\) is a bounded function and if \(\delta >0\) and \(r\in \mathbb{N}\), we define the $r$-th modulus of smoothness \(\omega _r(f,\delta )\) of \(f\) by
$$\omega _r(f,\delta )=\sup_{|h|\le \delta \\ x,x+rh\in I}
|\Delta _h^r f(x)|.$$
The \(r\)-th order modulus of smoothness of \(f\), \(r\in \mathbb{N}\), measured in \(L_p(I)\) spaces, \(p\ge 1\), is given by
$$\omega _r(f,\delta )_p=\sup_{0<h\le \delta }\|\Delta _h^r f\|_{L_p(I)},\ f\in L_p(I),\ \delta >0.$$
In the above \(\Delta _h^r f(x)=(E^h -I)^r f(x)\), \(E^h\) representing the translation operator.
For any \(k\le r\), \((E^h)^k f(x)=f(x+kh)\) if \(x\), \(x+kh\) belong to \(I\) and becomes null otherwise.
These moduli are the main tool in measuring the upper bound of the absolute error \(|(L_n f)(x)-f(x)|\), \(x\in I\), where \((L_n)_{n\ge 1}\) is a sequence of linear and positive operators.