Convergence of the Scaled Kernels

We are now ready to show that the sequence of order-$N$ scaled kernels converges to the infinite-order scaled kernel, within the whole support interval. Of course the convergence is guaranteed outside the support interval, since all the scaled kernels in the construction sequence are identically zero there. Starting from the $N=3$ scaled kernel shown in Figure 14, which is an everywhere continuous and differentiable function, we consider the action on it of the next first-order filter. Observe that within each one of the four intervals of length $\epsilon/2$ defined by the five invariant points this kernel is monotonic, and also that its first derivative is monotonic as well. The situation is as follows: in the first interval $[-\epsilon,-\epsilon/2]$ both the kernel function and its derivative are monotonically increasing; in the second interval $[-\epsilon/2,0]$ the kernel function is monotonically increasing, but its derivative is monotonically decreasing; in the third interval $[0,\epsilon/2]$, both the kernel function and its derivative are monotonically decreasing; in the fourth interval $[\epsilon/2,\epsilon]$, the kernel function is monotonically decreasing, but its derivative is monotonically increasing.

This means that this kernel function has a definite concavity in each of the four intervals. Let us now consider the action of the next instance of the first-order filter, at each point within the final support interval $[-\epsilon ,\epsilon ]$. We can say that either one of the five invariant points is contained within the support of the filter, or none is. If one of them is contained in the support, then we have already established that the support of the filter is contained within an interval where the kernel function is linear, and therefore the filter acts as the identity. In this case the kernel function is not changed at all. Otherwise, the support is contained within one of the four intervals where the kernel function has a definite concavity. In this case the kernel function will be changed, but its monotonic character, and that of its derivative, will be preserved. In other words the next kernel will have the same monotonicity and concavity properties on the same four intervals. Since this argument can then be iterated, we conclude that all subsequent kernel functions in the construction sequence have these same monotonicity and concavity properties, on the same four intervals.

Let us now consider the action of the first-order filter at any subsequent stage of the construction process. Once again, we have that either one of the five invariant points is contained within the support of the current filter, or none is. If one of the points is contained in the support, then the support of the filter is contained within an interval where the current kernel function is linear, and therefore the filter acts as the identity, so that the value of the kernel function is not changed. If none of the five points is contained within the support, then that support is contained within one of the four intervals where the current kernel has the same monotonicity and concavity properties of all the others in the sequence, starting with $N=3$. This means that at all stages of the construction process the points of the graph of the current kernel will always be changed in the same direction within these four intervals, being therefore always increased in the intervals $[-\epsilon,-\epsilon/2]$ and $[\epsilon/2,\epsilon]$, and always decreased in the intervals $[-\epsilon/2,0]$ and $[0,\epsilon/2]$.

What we may conclude from this is that, given any value of $\theta$ within $[-\epsilon ,\epsilon ]$, either it is one of the invariant points, at which all order-$N$ kernel functions have the same values, and therefore where the sequence of kernel functions converges, or it is a point strictly within one of the four intervals where the kernel functions have definite monotonicity and concavity properties. In this case the sequence of values of the order-$N$ kernel functions at that point form a monotonic real sequence. Since this is a monotonic sequence of real numbers that is bound from below by zero and from above by $1/\epsilon$, it follows that the sequence converges. Since we may therefore state that the point-wise convergence holds for all points within the final support interval $[-\epsilon ,\epsilon ]$, and recalling that outside this interval all order-$N$ kernels are identically zero, we conclude the the sequence of order-$N$ scales kernels converges, in the $N\to \infty$ limit, to the infinite-order scaled kernel, a definite limited real function $\bar{K}_{\epsilon}^{(\infty)}(\theta)$ with compact support.