Conclusions

We have shown that the real first-order low-pass filter, if taken in the $\epsilon \to 0$ limit, can be used to classify all possible generalized functions defined within a closed real interval into two disjoint classes, a class of combed generalized functions and a complementary class of ragged generalized functions. Although the first one contains, by and large, more regular members and less singular members than the second, the classification is not based simply on smoothness, since both classes do contain singular members. In addition to this, given any ragged function, there always is a corresponding combed function, from which it differs only by a zero-measure function. In this way, the concept suggests itself of the process of ``combing'' ragged functions. They can be ``combed'' so long as they can be characterized by a sequence of Taylor-Fourier coefficients, regardless of the convergence properties of the corresponding Fourier series. In all cases there are several ways in which this ``combing'' can be accomplished, including some that are algorithmically sound and useful.

We have also shown that the class of combed functions contains all those generalized functions that can be obtained as the $\rho\to 1$ limits of inner analytic functions. This same class is also that which contains the limiting generalized functions of all convergent Fourier series, as well as the limiting generalized functions obtained from all convergent center series built from the sequences of Fourier coefficients of real generalized functions. At this time we are not able to ascertain whether of not all combed generalized functions can be represented by inner analytic functions. This is so because although we have shown in [5] that the set of inner analytic functions includes representations of at least some locally non-integrable functions, we do not yet know whether or not it contains all of them. In other words, it is an open question whether or not all generalized functions which are locally integrable almost everywhere are the derivatives of some order of integrable generalized functions, and therefore can be represented by inner analytic functions in the way presented in [5], using the extended Taylor-Fourier coefficients introduced there.

It is interesting that, if one does not actually take the $\rho\to 1$ limit, but just gets sufficiently close to the unit circle to decrease sufficiently the errors involved, then one can draw the conclusion that, given any level of precision, there is an analytic real function that represents the physics within that precision. If one considers the restriction of the real part of any given inner analytic function to the circle of radius $\rho=1-\delta$, for some arbitrarily small but non-zero $\delta$, it becomes clear that this restriction is an analytic real function, since it is infinitely differentiable everywhere. If the limiting function represents some given physics within some error level and there are no hard singularities (where the representation must break down anyway), then the analytic real function obtained by this restriction with a sufficiently small $\delta$ represents the same physics as the function obtained in the $\rho\to 1$ limit, within the same error level. We may thus conclude that, given any real function that represents some physics within the finite and non-zero errors that are made inevitable by the fundamental physical limitations on the determination of all physical quantities that vary almost continuously, there is always a real analytic function arbitrarily close to it, that represents the same physics equally well, that is within the same level of precision.

The same conclusion can be reached if we start from the fact that the real functions that carry physical meaning in physics applications can also be represented by series in a basis of orthogonal functions, which is often the case when the functions are solutions of boundary value problems involving partial differential equations. Besides the Fourier basis of functions, this includes such commonly used bases as those formed by Legendre polynomials and those formed by Bessel functions of various types, among many others. One property that all these bases share is that they are infinite but discrete sets of analytic functions. If the series representing the functions are convergent, then the real functions can be represented by partial sums of the series within an arbitrarily given finite level of precision. However, these partial sums are finite sums of analytic functions and therefore are themselves analytic. In all these cases it follows that the physics can always be represented by analytic functions arbitrarily well, that is, within the finite and non-zero errors that are mandated by the fundamental physical limitations on the determination of all physical quantities that vary almost continuously.

As a final thought, one can pose the question of whether or not all solutions to linear partial differential equations are necessarily combed generalized functions. We believe that the right way to interpret this question is to ask whether or not there is an even more general way to extend the scope of differential equations that the transition from normal functions to generalized functions, and from normal derivatives to generalized derivatives, defined as we do here. So far as can be currently ascertained, this seems to be an open question.