Action of the Filter in Regions of Definite Concavity

Consider the action of the first-order filter in a region where the graph of $f(\theta)$ has definite concavity, turned in a definite direction, and within which the support interval of the filter fits. As shown in [4], if $f(\theta)$ happens to be a linear function within the support of the filter around a given point, then the filter is the identity and therefore $f_{\epsilon}(\theta)=f(\theta)$ at that point. On the other hand, if $f(\theta)$ is not a linear function and its concavity is turned down, them some values of the function within the support of the filter must be smaller that in the case of the linear function. Since the filtered function $f_{\epsilon}(\theta)$ is defined as an average of the values of $f(\theta)$, it follows that, if the concavity of the graph of $f(\theta)$ is turned down, then

$$f_{\epsilon}(\theta) < f(\theta).$$

In the same way, we may conclude that if the concavity of the graph of $f(\theta)$ is turned up, then

$$f_{\epsilon}(\theta) > f(\theta).$$

In other words, in regions where the function $f(\theta)$ has its concavity turned in a definite direction, and within which the support interval of the filter fits, the action of the filter always changes the value of the function in the direction to which its concavity is turned.