Properties of Combed Functions

Although the class of combed function includes many singular objects such as the delta ``function'' and at least some locally non-integrable functions, the generalized functions within the set have a significant collection of common properties that considerably simplifies their handling. Note that this is a large set of objects including many singular ones of common use in physics, since combed function can be non-differentiable, discontinuous, and may diverge to infinity at singular points, which may or may not be integrable ones. For the purposes of this section let us limit ourselves to the case in which the combed generalized functions are integrable. This will make it easier for us to list their main common properties, most of which have been demonstrated before. The detailed extension of these properties to the complete set of combed generalized functions is therefore postponed to some future opportunity.

• To start with, combed functions have no finite-spike singularities. In fact, any pathologies that do not have a definite non-zero effect on the integral of the function, such as finite discontinuities on a zero-measure subset of the domain, are simply eliminated by the application of the low-pass filter.

• Since it is the $\epsilon \to 0$ limit of a filtered function, every combed function assumes at every given point the value given by the average of the two lateral limits of the original function to that point, so long as these limits exist,

  
  

  $$\frac{{\cal L}_{\oplus}+{\cal L}_{\ominus}}{2},$$
  
  

  where

  $$\begin{eqnarray*} {\cal L}_{\oplus} & = & \hspace{-1em} \lim_{\hspace{0.8em}... ...pace{-1em} \lim_{\hspace{0.8em}\theta\to\theta_{0-}}f(\theta), \end{eqnarray*}$$
  
  
  
  

  which in particular holds at all isolated discontinuities where the two lateral limits exist. Note that, since in the neighborhoods at the two sides of $\theta_{0}$, where $f(\theta)$ is continuous, the $\epsilon \to 0$ limit of $f_{\epsilon}(\theta)$ reproduces $f(\theta)$, these two lateral limits of $f(\theta)$ coincide with the two corresponding lateral limits of $f_{\epsilon}(\theta)$ in the $\epsilon \to 0$ limit.

• Every combed function $f(\theta)$ has a well-defined and unique sequence of Taylor-Fourier coefficients, from which it can be recovered strictly everywhere within its domain of definition. From the sequence of Taylor-Fourier coefficients one can always construct the corresponding inner analytic function $w(z)$. The combed real function is always given by the $\rho\to 1$ limit of the real part of its corresponding inner analytic function,

  
  

  $$f(\theta) = \lim_{\rho\to 1}\Re[w(z)],$$
  
  

  everywhere within that domain, even if the Fourier series of that real function diverges everywhere.

• If the Fourier series of a combed function converges at all, then it converges to that combed function everywhere in its domain of definition.

• If the Fourier series of a combed function diverges, and so long as there is at most a finite number of dominant singularities of the corresponding inner analytic function on the unit circle, as they were defined in [2], it is possible to devise other expressions involving trigonometric series that converge to the real function, and that do so as fast as one may wish. We call these alternative trigonometric series ``center series'', and the algorithm to construct them is explained in [2].

• Every combed function is the smoothest member of the class of zero-measure equivalent functions that it belongs to, as was discussed in Appendix C of [4]. Given another member of that class, which is therefore a ragged function, there is no difficulty in ``combing'' it, that is in finding the combed function from which it differs only by a zero-measure function.

• A ragged function can be combed by any one of the following methods: application to it of the low-pass filter in the $\epsilon \to 0$ limit; construction of its Fourier series, if convergent, and therefore of the limiting function of that series; construction of an expression containing an alternative trigonometric series, if the Fourier series fails to converge, and therefore of the limiting function of that alternative series; construction of its inner-analytic function and therefore of the limit of the real part of that complex function to the unit circle.

In any given setting, if one limits the set of relevant real functions to be discussed to combed generalized functions, then the use of the very characteristic ``almost everywhere'' arguments in the harmonic analysis of the functions become more sparse and better focused on the relevant features of the functions. The possible exceptional points of the ``almost everywhere'' arguments become only those singular points on the unit circle that are brought about by the construction of the corresponding inner analytic functions.