Introduction

In a recent series of papers [1,2,3,4,5] we established a correspondence between, on the one hand, real functions and generalized real functions or distributions, all defined within a compact interval, and on the other hand, certain analytic functions defined within the open unit disk of the complex plane [6]. This correspondence involves in a central way the Fourier coefficients of the real functions, as well as the issue of the representability of the generalized functions by their sequences of Fourier coefficients [7]. The generalized functions as defined in [5] are to be understood loosely in the spirit of the Schwartz theory of distributions [8]. The new correspondence that was established allows one to deal with a large set of generalized functions, either singular or not, via their representation in terms of analytic functions, and therefore through the use of very solidly established analytic procedures.

In a related paper [9] we introduced a set of linear low-pass filters as tools that can be used to deal efficiently with divergent or poorly convergent Fourier series resulting from the resolution of boundary value problems of partial differential equations. In one of the papers [3] of the series mentioned above these filters were integrated into the structure of the aforementioned correspondence between real generalized functions and complex analytic functions. This was done via the introduction of complex low-pass filters within the open unit disk of the complex plane, acting on complex analytic functions, that reproduce the action of the real low-pass filters on the real functions when one takes the limit from within the open unit disk to the unit circle.

Here we will use these elements to show that the first-order linear low-pass filter can be used to establish a useful classification of all the generalized functions. This will separate the set of all generalized functions that one can define on the unit circle into two disjoint subsets. One of these we will call the set of combed functions, the other we will call the set of ragged functions. Although most of the more profoundly pathological generalized functions are included in the second subset, the classification is not based simply on smoothness, since the Dirac delta ``function'' and many other singular generalized functions, as well as many singular normal functions, are in fact classified as combed functions.

We will argue that the set of combed generalized functions is sufficient for all the needs of physics, in the role of tools for the description of the observable aspects of nature. It should be pointed out that, while one does not need real function in the continuum to describe aspects of nature that are intrinsically discrete, such as the spin of elementary particles, real generalized functions in the continuum can be advantageously used to describe the behavior of physical quantities that depend on variables that vary almost continuously, such as spatial positions. They can also be used as very good approximations to describe those physical systems that involve an extremely large number of degrees of freedom, such as extended material objects. The argument is that combed generalized functions suffice for these roles. The universe of applicability includes the whole of classical physics, as well as all the observable quantities that vary almost continuously in quantum mechanics and quantum field theory.