Invariance of the Sign of the Second Derivative

If we assume that $f(\theta)$ is differentiable in the domain where the filter is applied, then $f_{\epsilon}(\theta)$ can be differentiated twice, and we may obtain its second derivative by simply differentiating once Equation (8), which results in

$$\frac{d^{2}f_{\epsilon}(\theta)}{d\theta^{2}} = \frac{1}{... ... - \frac{df}{d\theta}\!\left(\theta-\epsilon\right) \right].$$

This is valid so long as the support interval of the filter fits completely inside the region where $f(\theta)$ is differentiable. This immediately implies that, in a region where the derivative of $f(\theta)$ increases monotonically we have

$$\begin{eqnarray*} \frac{df}{d\theta}\!\left(\theta+\epsilon\right) & \geq & \... ...\\ \frac{d^{2}f_{\epsilon}(\theta)}{d\theta^{2}} & \geq & 0. \end{eqnarray*}$$

We may therefore conclude that the derivative of $f_{\epsilon}(\theta)$ also increases monotonically within the sub-region where the support interval of the filter fits inside the region in which $f(\theta)$ is differentiable. In the same way, in a region where the derivative of $f(\theta)$ decreases monotonically we have

$$\begin{eqnarray*} \frac{df}{d\theta}\!\left(\theta+\epsilon\right) & \leq & \... ...\\ \frac{d^{2}f_{\epsilon}(\theta)}{d\theta^{2}} & \leq & 0. \end{eqnarray*}$$

We may therefore conclude that the derivative of $f_{\epsilon}(\theta)$ also decreases monotonically within that same sub-region. In other words, the monotonic character of the variation of the derivative of a function is invariant by the action of the filter. In particular, at points where $f(\theta)$ is twice differentiable the sign of its second derivative is invariant by the action of the filter.

This implies that in regions where the second derivative of $f(\theta)$ has constant sign, and therefore the concavity of its graph is turned in a definite direction, up or down, the action of the filter keeps that concavity turned in the same direction. In other words, away from inflection points, in regions where the graph of $f(\theta)$ has a definite concavity turned in a definite direction, $f_{\epsilon}(\theta)$ has the same concavity, turned in the same direction.