Partial Differential Equations

When Fourier series are used for applications in physics, involving
partial differential equations, sometimes the process of resolution
results in divergent series for some quantities. In this paper we argue
that the use of linear low-pass filters is a valid way to regularize
such divergent series. In particular, we show that these divergences are
always the result of oversimplification in the proposition of the
problems, and do not have any fundamental physical significance. We
define the first-order linear low-pass filter in precise mathematical
terms, establish some of its properties, and then use it to construct
higher-order filters. We also show that the first-order linear low-pass
filter, understood as a linear integral operator in the space of real
functions, commutes with the second-derivative operator. This can
greatly simplify the use of these filters in physics applications, and
we give a few simple examples to illustrate this fact.

- Introduction
- The Low-Pass Filters on the Real Line

- Application in Partial Differential Equations
- Conclusions
- Acknowledgements
- Appendix: Properties of the First-Order
Filter
- Invariance of Linear Functions
- Action on Powers and Polynomials
- Differentiability of Filtered Functions
- Continuity of Filtered Functions
- Action on Dirac's Delta ``Function''
- Reduction to the Identity
- Points of Discontinuity
- Points of Non-Differentiability
- Invariance of Definite Integrals
- Periodicity of Filtered Functions
- Invariance of Averages Over the Period
- Action on the Fourier Coefficients with
- Completeness of the Set of Eigenfunctions

- Appendix: Examples of Use of the First-Order
Filter

- Bibliography