The Classification

The first-order low-pass filter can now be used to define a classification of generalized functions. In order for the filter to be applicable, and therefore for the classification to be feasible, the generalized functions must be locally integrable almost everywhere, but they do not have to be integrable on the whole domain. Observe, moreover, that we do not have to assume that the generalized functions which we start with in this argument were defined as $\rho\to 1$ limits of corresponding inner analytic functions. Therefore, given a generalized function $f(\theta)$ which is locally integrable almost everywhere on the unit circle, irrespective of whether or not it was defined as the $\rho\to 1$ limit of an inner analytic function, we say that it is a combed function if it is true that

$$f(\theta) = \lim_{\epsilon\to 0} f_{\epsilon}(\theta),$$ (5)

where this recovery of the original function from the $\epsilon \to 0$ limit of the filtered function holds everywhere. Otherwise we say that the generalized function is a ragged function. The results obtained in [9] and discussed in the previous section now imply that any continuous function is a combed function. We may consider adopting a corresponding complex definition, using the corresponding criterion for the inner analytic functions within the open unit disk. Therefore we may classify an inner analytic function as combed if it satisfies the condition that

$$w(z) = \lim_{\epsilon\to 0} w_{\epsilon}(z),$$ (6)

where the recovery of the original function from the $\epsilon \to 0$ limit of the filtered function holds everywhere within the open unit disk. However, we showed in the previous section that this is the case for all inner analytic functions. This means that every inner analytic function is combed within the open unit disk.

Since the complex low-pass filter reduces to the real low-pass filter on the unit circle, it follows at once that all the generalized functions obtained as limits to the unit circle of the real parts of inner analytic functions within the open unit disk are combed functions, which we will refer to as ``combed generalized functions''. While this simply reproduces the previously obtained results [9] for the case of continuous real functions on the unit circle, it also means that the Dirac delta ``function'' is a combed generalized function, which is quite a remarkable fact. In this particular case it is not difficult to verify this fact directly, but from the same result expressed by Equation (6) it follows that this is also true for all the derivatives of the delta ``function'', of all orders, which is not so simple and immediately apparent a fact.

Here is a simple direct proof that the delta ``function'' is a combed generalized function. Consider a delta ``function'' $\delta(\theta-\theta_{0})$ centered at $\theta_{0}$. It is not difficult to verify by direct calculation that the real filter with parameter $\epsilon$, if applied to this generalized function, produces a unit-integral rectangular pulse of width $2\epsilon$ and height $1/(2\epsilon)$ centered at that same point. As one decreases $\epsilon$, the $\epsilon \to 0$ limit of this one-parameter family of rectangular pulses is one of the many ways commonly used as a definition of the delta ``function''. Therefore one recovers the delta ``function'' in the $\epsilon \to 0$ limit, thus showing that the delta ``function'' is a combed generalized function.

A similar argument can be constructed for the first derivative of the delta ``function'', in which one must use some of the properties of the delta ``function'' itself, and the fact that the derivative of the integration kernel $K_{\epsilon}\!\left(\theta-\theta'\right)$ contains a pair of delta ``functions''. However, in this case it is much simpler to establish that the derivative of the delta ``function'' is a combed generalized function by simply relying on the fact that it is the $\rho\to 1$ limit of an inner analytic function, as was shown in [5]. The same argument is valid, with equal ease, for all the derivatives of the delta ``function'', of arbitrarily high orders.

It is also easy to see by direct calculation that any function with a few isolated step-type discontinuities where the two lateral limits exist, and which is otherwise continuous, is a combed function, so long as at the points of discontinuity the function is defined as the average of the two lateral limits. Note that this is the value to which the Fourier series of the function converges, if that series is convergent at all. Once again, this result can be easily derived from the fact that functions such as the ones just described can be obtained as the $\rho\to 1$ limits of the real parts of the corresponding inner analytic functions.