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.

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.