Products of Distributions

In the Schwartz theory of distributions one important theorem states that it is not possible to define, in a general way, the product of two distributions [#!NMSchwartz!#], which has the effect that the space of Schwartz distributions cannot be promoted from a vector space to an algebra. In this section we will interpret this important fact in the context of the representation of integrable real functions and singular distributions in terms of inner analytic functions. We start by noting that, although it is always possible to define the product of two inner analytic functions, which is always an inner analytic function itself, this does not correspond to the product of the two corresponding real functions or related objects. If we have two inner analytic functions given by


$\displaystyle w_{1}(z)$ $\textstyle =$ $\displaystyle u_{1}(\rho,\theta)
+
\mbox{\boldmath$\imath$}
v_{1}(\rho,\theta),$  
$\displaystyle w_{2}(z)$ $\textstyle =$ $\displaystyle u_{2}(\rho,\theta)
+
\mbox{\boldmath$\imath$}
v_{2}(\rho,\theta),$ (70)

the product of the two inner analytic functions is given by


$\displaystyle w(z)$ $\textstyle =$ $\displaystyle \left[
u_{1}(\rho,\theta)
u_{2}(\rho,\theta)
-
v_{1}(\rho,\theta)
v_{2}(\rho,\theta)
\right]$  
    $\displaystyle +
\mbox{\boldmath$\imath$}
\left[
u_{1}(\rho,\theta)
v_{2}(\rho,\theta)
+
v_{1}(\rho,\theta)
u_{2}(\rho,\theta)
\right],$ (71)

whose real part is not just the product $u_{1}(\rho,\theta)u_{2}(\rho,\theta)$. In fact, the problem of finding an inner analytic function whose real part is this quantity often has no solution. One can see this very simply by observing that both $u_{1}(\rho,\theta)$ and $u_{2}(\rho,\theta)$ are always harmonic functions, and that the product of two harmonic functions in general is not a harmonic function. Since the real and imaginary parts of an inner analytic function are always harmonic functions, it follows that the problem posed in this way cannot be solved in general. The only simple case in which we can see that the problem has a solution is that in which one of the two functions being multiplied is a constant function.

Let us state in a general way the problem of the definition of the product of two distributions. Suppose that we have two inner analytic functions such as those in Equation (74). The two corresponding real objects are $u_{1}(1,\theta)$ and $u_{2}(1,\theta)$, and their product, assuming that it can be defined in strictly real terms, is simply the real object $u_{1}(1,\theta)u_{2}(1,\theta)$. The problem of finding an inner analytic function that corresponds to this product is the problem of finding an harmonic function $u_{\pi}(\rho,\theta)$ whose limit to the unit circle results in this real object,


\begin{displaymath}
u_{\pi}(1,\theta)
=
u_{1}(1,\theta)u_{2}(1,\theta).
\end{displaymath} (72)

If one can find such a harmonic function, then it is always possible to find its harmonic conjugate function $v_{\pi}(\rho,\theta)$ and therefore to determine the inner analytic function


\begin{displaymath}
w_{\pi}(z)
=
u_{\pi}(\rho,\theta)
+
\mbox{\boldmath$\imath$}
v_{\pi}(\rho,\theta),
\end{displaymath} (73)

which corresponds to the product of the two real objects. According to the construction presented in [#!CAoRFI!#], this can always be done so long as the product $u_{1}(1,\theta)u_{2}(1,\theta)$ is an integrable real function of $\theta$. However, if $u_{1}(1,\theta)$ and $u_{2}(1,\theta)$ are singular objects that can only be defined as limits from within the open unit disk, then the product may not be definable in strictly real terms, and it may not be possible to find an inner analytic function such that the $\rho\to 1_{(-)}$ limit of its real part results in this product, interpreted in some consistent way. This is the content of the theorem that states that this cannot be done in general.

It is not too difficult to give examples of products which are not well defined. It suffices to consider the product of any two singular distributions which have their singularities at the same point on the unit circle. If one considers integrating the resulting object and using for this purpose the fundamental property of any of the two distributions involved, one can see that the integral is not well defined in the context of the definitions given here for the singular distributions. Although one cannot rule out that some other definition can be found to include some such cases, we certainly do not have one at this time.

We thus see that we are in fact unable to promote the whole space of integrable real functions and singular distributions to an algebra. However, there are some sub-spaces within which this can be done. Under some circumstances one can solve the problem of defining within the complex-analytic structure the product of two integrable real functions. This cannot be done for the whole sub-space of integrable real functions, because there is the possibility that the product of two integrable real function will not be integrable. However, if we restrict the sub-space to those integrable real functions which are also limited, then it can be done. This is so because the product of two limited integrable real functions is also a limited real function, and therefore integrable. In this way, one can find the inner analytic function that corresponds to the product, since according to the construction which was presented in [#!CAoRFI!#], this can be done for any integrable real function. The resulting inner analytic function will not, however, be related in a simple way to the inner analytic functions of the two factor functions.

One case in which the product can always be defined is that of an integrable real function with a Dirac delta ``function'', so long as the real function is well defined at the singular point of the delta ``function''. Given the nature of the delta ``function'', this is equivalent to multiplying it by a mere real number, the value of the integrable real function at the singular point of the delta ``function'',


\begin{displaymath}
g(\theta)\delta(\theta-\theta_{1})
=
g(\theta_{1})\delta(\theta-\theta_{1}).
\end{displaymath} (74)

The corresponding inner analytic function is therefore given simply by $g(\theta_{1})w_{\delta}(z,z_{1})$. Similar statements are true, of course, for all the derivatives of the delta ``function''. Therefore, in all such cases there is no difficulty in determining the inner analytic function that corresponds to the product.

Note that this difficulty relates only to the definition of the product of two real objects on the unit circle. As was observed before, for all the singular distributions their definition by means of inner analytic functions always provides the means to determine whether or not they can be applied to a given real object, so long as it is represented by an inner analytic function, and determines what results from that operation, if it is possible at all.