Proof of Convergence to the Infinite-Order Kernel

In this appendix we will offer proof of the convergence of the sequence of order-$N$ scaled kernels to the infinite-order scaled kernel, in the $N\to \infty$ limit. The point is to show that the infinite sequence of real functions $\bar{K}_{\epsilon_{N}}^{(N)}(\theta)$, with $\epsilon<\pi$, converges in the $N\to \infty$ limit to a definite regular real function with finite support within $[-\epsilon ,\epsilon ]$, which we denote as $\bar{K}_{\epsilon}^{(\infty)}(\theta)$. In addition to this, we will establish that the infinite-order scaled kernel $\bar{K}_{\epsilon}^{(\infty)}(\theta)$ is a $C^{\infty}$ function, and that it is not an analytic function in the real sense of the term.

In order to do this we must first establish a few more properties of the first-order filter, in addition to those demonstrated in [3]. We will also have to examine more closely some aspects of the process of construction of the infinite-order scaled kernel.