Invariant Points of the Scaled Kernels

Figure 13: The scaled kernel for $\epsilon =0.5$ and $N=2$, plotted as a function of $\theta$ over the support interval $[-\epsilon ,\epsilon ]$. The five invariant points are marked by stars, and the support interval of the next filter in the construction sequence is shown. The dashed lines mark the four intervals defined by the five invariant points. The dotted lines mark the sectors where the function is defined in a piece-wise fashion.

Let us now show that there are five points of the graphs of the order-$N$ scaled kernels that remain invariant throughout the construction of the infinite-order scaled kernel. These are the following: the point of maximum at $\theta=0$, where the value of the kernel function is $1/\epsilon$; the two points of minimum at $\theta=\pm\epsilon$, where the value is zero; and the two inflection points at $\theta=\pm\epsilon/2$, where the value is $1/(2\epsilon)$. These five points are marked with stars on the graphs in Figures 12, 13 and 14, which display the first three kernels in the construction sequence. Note that in the case of the discontinuous $N=1$ kernel in Figure 12 we choose the values at the points of discontinuity according to the criterion of the average of the lateral limits, which defines these two future points of inflection in a way that is compatible with this invariance.

Note now that in Figure 12 the points of maximum and minimum are within sectors where the kernel function is linear, in intervals of size $\epsilon$ (or more, in the case of the points of minimum) around these points. The support of the next filter to be applied, in the sequence leading to the construction of the infinite-order scaled kernel, is also shown in the graph. Since this support has length $\epsilon/2$, it fits into the intervals where the kernel function is linear, in which case it acts as the identity, according to one of the properties of the first-order filter [4]. Therefore, after the application of this next filter intervals of length $\epsilon/2$ will remain around these three points, where the next kernel function so generated is still linear.

Figure 14: The scaled kernel for $\epsilon =0.5$ and $N=3$, plotted as a function of $\theta$ over the support interval $[-\epsilon ,\epsilon ]$. The five invariant points are marked by stars, and the support interval of the next filter in the construction sequence is shown. The dashed lines mark the four intervals defined by the five invariant points. The dotted lines mark the sectors where the function is defined in a piece-wise fashion.

The result of this operation, which is the $N=2$ scaled kernel, is shown in Figure 13. Note that in this case all the five points listed before have around them intervals of length $\epsilon/2$ where the kernel function is linear. Once more the support of the next filter to be applied, in the sequence leading to the construction of the infinite-order scaled kernel, is shown in the graph. Since this support has length $\epsilon/4$, it fits into the intervals where the kernel function is linear, in which case it acts as the identity, so that after its application intervals of length $\epsilon/4$ will remain around all these five points, where the next kernel function generated is still linear. In particular, this will keep the points invariant, since they are within sectors where the kernel functions are linear and hence where the first-order filter acts as the identity.

The result of this last operation, which is the $N=3$ scaled kernel, is shown in Figure 14. Once again all five points are within sectors of length $\epsilon/4$ where the kernel function is linear. Since the support of the next filter to be applied has now length $\epsilon/8$, once more it will keep these points invariant. It is now quite clear that both the length of the linear sectors and the length of the support of the next filter will be scaled down exponentially during the process of construction of the infinite-order kernel, with the support being always half the length of the intervals, and therefore fitting within them. This establishes that the five points we listed here are in fact invariant throughout the construction of the infinite-order scaled kernel. In particular, it follows that these are the values of the infinite-order scaled kernel function at these five points, and that the sequence of order-$N$ scaled kernel functions in fact converges to the infinite-order scaled kernel function at these five points.