Conclusions

Linear low-pass filters of arbitrary orders can be easily and elegantly defined on the complex plane, within the unit disk, acting on inner analytic functions. Within the open unit disk the filter simply maps inner analytic functions onto other inner analytic functions. Through the correspondence of these functions with FC pairs of DP Fourier series, these filters reproduce the linear low-pass filters that were defined in a previous paper, acting on the corresponding DP real functions defined on the unit circle. Several of the properties of these filters are then clearly in view, given the known properties of that correspondence.

The effect of the first-order low-pass filter, as seen in the complex plane, is characterized as a process of singularity splitting, in which each singularity of an inner analytic function on the unit circle is exchanged for two softer singularities over that same circle. This has the effect of improving the convergence characteristics of the DP Fourier series, and also of rendering the corresponding DP real functions smoother after the filtering process. Higher-order filters correspond to the iteration of this process on the unit circle, producing ever larger collections of ever softer singularities on that circle.

A discussion of the problems encountered when one tries to define an infinite-order low-pass filter acting on real functions, in the most immediate way, led to the detailed construction of such an infinite-order filter, within a compact support. The representation of the filters in the complex plane was instrumental for the success of this construction. The infinite-order filter is defined in terms of an infinite-order scaled kernel, with compact support given by a real parameter $\epsilon$, which can be as small as one wishes.

The infinite-order scaled kernel is defined as the limit of a sequence of order-$N$ scaled kernels, and proof of the convergence of the sequence was presented. It was also shown that the infinite-order scaled kernel is a $C^{\infty}$ real function, but not an analytic real function. The infinite-order scaled kernel can be given in a fairly explicit way as a limit of a Fourier series, which converges extremely fast. Once the infinite-order filter is defined in the periodic interval, it is a simple matter to define a corresponding infinite-order filter that acts on the whole real line.

This infinite-order scaled filter, acting on any merely integrable real function on the unit circle, has as its result a real function that is $C^{\infty}$ on the unit circle, while making on the original function only changers with the finite range $\epsilon$. The same is true for real function defined on the whole real line. Therefore, one obtains as a result of this construction a tool that can produce from any integrable real function corresponding $C^{\infty}$ functions, making changes only within a range $\epsilon$ that can be as small as desired.

This allows us to use these filters in physics applications, if we use values of $\epsilon$ sufficiently small in order not to change the description of the physics within the physically relevant scales of any given problem. By reducing the value of $\epsilon$ these $C^{\infty}$ functions can be made as close as one wishes to the corresponding original functions, according to a criterion that has a clear physical meaning, as explained in a previous paper.

In addition to this, the construction of the filter is equivalent to proof that there are many real functions that are $C^{\infty}$ but that are not analytic, and that are typically not extensible analytically to the complex plane. The filter can be used to produce examples of such functions in copious quantities. It is quite easy to obtain accurate values for the filtered functions by numerical means, and thus to represent the action of the filter in practical applications.