Conclusions

Linear low-pass filters of arbitrary orders can be easily defined on the real line, in a very simple way, either on the whole line or within a periodic interval. We presented a definition of such filters in precise mathematical terms, and also wrote them as linear integral operators acting in the space of integrable real functions, expressed as integrals involving certain kernel functions. We established several of the main properties of the first-order filter. Due to the linearity of the filters some of these properties, those involving the concept of invariance, are immediately generalizable to the higher-order filters.

The use of the filters on divergent Fourier series produces other series which are convergent, but which remain closely related to the original problem within the physics application being dealt with, so long as the range $\varepsilon$ is sufficiently small. It also produces series that converge faster to smoother functions, and that can be differentiated a certain number of times, as required by the applications involved, without resulting in divergent series. We thus acquire a useful set of tools to deal with Fourier series in a way that has a clear physical meaning in the context of applications in physics. This set of tools can then be used as a probe into the physical structure of the problems being dealt with.

It also follows in a very simple way that the filter operators commute with the second derivative operator. We showed this in the case of the first-order filter, and since the higher-order filters are just the first-order one applied successive times, the result is immediately extended to them. This leads to the fact that in many cases one may obtain the solutions of the filtered boundary valuer problems by the simple application of the filters directly to the solutions of the unfiltered problems. This is very easily done when the solutions are expressed as Fourier series, and may greatly simplify matters in practice.

Since any limited integrable function will have a limited set of Fourier coefficients, it suffices to apply to such functions the first-order linear low-pass filter of range $\varepsilon$ twice, or the second-order filter of range $\varepsilon$ just once, in order to ensure the absolute and uniform convergence of the resulting Fourier series. Therefore, it becomes clear that any divergence of the original series must be due to detailed structure that exists below the length scale that characterizes these filters. For small enough values of the range of the filters such structure cannot have a bearing on the physics involved, which allows us to use the filters in this way. But besides this practical application the filters give us a simple, clear and intuitive way to understand the origin of the eventual divergences of the Fourier series.