I am currently reviewing a paper published in the Journal of Approximation Theory and working on proving some unresolved properties. Below are the necessary definitions and the properties themselves:
The averaged modulus of smoothness:
In order to establish approximation results for the NN operators in the above context, with respect to the $p$-norm, we need to introduce the notion of the so-called averaged modulus of smoothness.
Let $f$ be a function defined and bounded on $I=[0,1]$.
We now define the so-called local modulus of smoothness of the function $f$ of order $k \in \mathbb{N}(k \geq 1)$ at a point $x \in I$ as follows:
\begin{equation*}\omega_{k}(f, x ; \delta):=\sup \left\{\left|\Delta_{h}^{k} f(t)\right|: t, t+h k \in\left[x-\frac{k \delta}{2}, x+\frac{k \delta}{2}\right] \cap I\right\} \end{equation*}
where $0<\delta \leq 2 / k$, and
$$\Delta_{h}^{k} f(t):=\sum_{j=0}^{k}(-1)^{j+k}\binom{k}{j} f(t+j h)$$
Notice that, $\omega_{k}(f, x ; \delta)$ as a function of the variable $x$ is defined for every $x \in I$, and obviously, it turns out that:
\begin{equation*}\omega_{k}(f, \delta)=\left\|\omega_{k}(f, \cdot ; \delta)\right\|_{\infty} \tag{14}\end{equation*}
where $\omega_{k}(f, \delta)$ denotes the usual (uniform) modulus of smoothness of order $k$ .
The introduction of the above-considered averaged modulus of smoothness can be obtained from the local moduli using integral norms in (14) instead of the uniform norm, as we now show.
By the symbol $M(I)$ we now denote the set of all bounded and measurable functions $f: I \rightarrow \mathbb{R}$. We can state the following.
Definition 3.1. The averaged modulus of smoothness of order $k$ (or $\tau$-modulus) of a function $f \in M(I)$ is the following function of $0<\delta \leq 2 / k$ :
$$\tau_{k}(f, \delta)_{p}:=\left\|\omega_{k}(f, \cdot ; \delta)\right\|_{p}=\left\{\int_{-1}^{1}\left[\omega_{k}(f, x ; \delta)\right]^{p} d x\right\}^{1 / p}, \quad 1 \leq p<+\infty$$
The tool introduced in Definition 3.1 represents a mean of the local modulus of smoothness.
The above definition of averaged modulus of smoothness is due to Sendov and Korovkin; originally, it has been introduced in a slightly different form, and only subsequently it has been formulated as in Definition 3.1. The main peculiarity of $\tau_{k}$ is that, it allows to evaluate the increments of a given function $f$ at any fixed point $x$ in $I$, differently to the usual $L^{p}$ moduli of smoothness that are not able to consider the increments of $f$ on subsets of $I$ with null measure.
The $\tau$-modulus has five basic properties, which are analogous to the basic properties of the usual moduli of smoothness.
(a) monotonicity:
$$\tau_{k}\left(f, \delta^{\prime}\right)_{p} \leq \tau_{k}\left(f, \delta^{\prime \prime}\right)_{p}, \quad \text { for } \quad \delta^{\prime} \leq \delta^{\prime \prime}$$
(b) semi-additivity:
$$\tau_{k}(f+g, \delta)_{p} \leq \tau_{k}(f, \delta)_{p}+\tau_{k}(g, \delta)_{p}, \quad \delta>0$$
(c) a higher order modulus can be estimated by modulus of lower order, i.e.:
$$\tau_{k}(f, \delta)_{p} \leq 2 \tau_{k-1}\left(f, \frac{k}{k-1} \delta\right)_{p}, \quad \delta>0$$
(d) the modulus of order $k$ of $f$ can be estimated by the modulus of order $k-1$ of $f^{\prime}$ whenever it exists:
$$\tau_{k}(f, \delta)_{p} \leq \delta \tau_{k-1}\left(f^{\prime}, \frac{k}{k-1} \delta\right)_{p}, \quad \delta>0$$
(e) the following inequality holds:
\begin{equation*}\tau_{k}(f, n \delta)_{p} \leq(2 n)^{k+1} \tau_{k}(f, \delta)_{p}, \quad n \in \mathbb{N}, \delta>0 \end{equation*}
Can anyone provide me hints to prove above properties?.