Piecewise Polynomial Functions

It is important to note that the Dirac delta ``function'' and all its derivatives, with their singularities located at a given point $z_{1}$ on the unit circle, are all contained within a single integral-differential chain, making up, in fact, only a part of that chain, the semi-infinite chain starting from the delta ``function'' and propagating indefinitely in the differentiation direction along the chain. However, the chain propagates to infinity in both directions. In order to complete its analysis, we must now determine what is the character of the real objects in the remaining part of that chain, in the integration direction. In fact, they are just integrable real functions, although they do have a specific character. They consist of sections of polynomials wrapped around the unit circle, of progressively higher orders, and progressively smoother across the singular point, as functions of $\theta$, as one goes along the integral-differential chain in the integration direction.

Let us illustrate this fact with a few simple examples. Instead of performing angular integrations of the inner analytic functions, we will do this by performing integrations on the unit circle. As was established in [#!CAoRFI!#], one can determine these functions by simple integration on $\theta$, so long as one remembers two things: first, to make sure that the real functions or related objects to be integrated on $\theta$ have zero average over the unit circle, and second, to choose the integration constant so that the resulting real functions also have zero average over the unit circle. For example, the integral of the zero-average delta ``function''

$$\delta^{0\prime}(\theta-\theta_{1}) = \delta(\theta-\theta_{1}) - \frac{1}{2\pi},$$ (61)

which is obtained from the real part of the proper inner analytic function in Equation (33), can be integrated by means of the simple use of the fundamental property of the delta ``function'', thus yielding

$$<tex2html_comment_mark>\renewedcommand{arraystretch}{2.0} ... ...laystyle 2\pi}} & \mbox{for} & \Delta\theta<0, \end{array}$$

where $\Delta\theta=\theta-\theta_{1}$. This is a sectionally linear function, with a single section consisting of the intervals $[-\pi,0)$ and $(0,\pi]$, thus excluding the point $\Delta\theta=0$ where the singularity lies, and with a unit-height step discontinuity at that point. Note that it is an odd function of $\Delta\theta$. The next case can now be calculated by straightforward integration, which yields

$$<tex2html_comment_mark>\renewedcommand{arraystretch}{2.0} ... ...laystyle 4\pi}} & \mbox{for} & \Delta\theta<0. \end{array}$$

This is a sectionally quadratic function, this time a continuous function, again with the same single section excluding the point $\Delta\theta=0$, but now with a point of non-differentiability there. Note that it is an even function of $\Delta\theta$. The next case yields, once more by straightforward integration,

$$<tex2html_comment_mark>\renewedcommand{arraystretch}{2.0} ... ...aystyle 12\pi}} & \mbox{for} & \Delta\theta<0. \end{array}$$

This is a sectionally cubic continuous and differentiable function, again with the same single section excluding the point $\Delta\theta=0$. Note that it is an odd function of $\Delta\theta$. The trend is now quite clear. All the real functions in the chain, in the integration direction starting from the delta ``function'', are what we may call piecewise polynomials, even if we have just a single piece within a single section of the unit circle, as is the case here. The $n^{\rm th}$ integral is a piecewise polynomial of order $n$, which has zero average over the unit circle, and which becomes progressively smoother across the singular point as one goes along the integral-differential chain in the integration direction.

In order to generalize this analysis, we must now consider linear superpositions of delta ``functions'' and derivatives of delta ``functions'', with their singularities situated at various points on the unit circle. A simple example of such a superposition, which we may use to illustrate what happens when we make one, is that of two delta ``functions'', with singularities at $\theta=0$ and at $\theta=\pm\pi$, added together with opposite signs,

$$f(\theta) = \delta(\theta) - \delta(\theta-\pi),$$ (62)

that corresponds to the following inner analytic function, which this time is already a proper inner analytic function, with two simple poles at $z=\pm 1$,

$$w(z) = -\, \frac{1}{\pi}\, \frac{z}{z-1} + \frac{1}{\pi}\, \frac{z}{z+1}$$

$$= -\, \frac{2}{\pi}\, \frac{z}{z^{2}-1}.$$ (63)

Since we have now two singular points, one at $z=1$ and another at $z=-1$, corresponding respectively to $\theta=0$ and $\theta=\pm\pi$, we have now two sections, one in $(-\pi,0)$ and another in $(0,\pi)$. The inner analytic functions at the integration side of the integral-differential chain to which this function belongs are obtained by simply adding the corresponding inner analytic functions at the integration sides of the integral-differential chains of the two functions that are superposed. The same is true for the corresponding real objects within each section of the unit circle. Since the real functions corresponding to each one of the two delta ``functions'' that were superposed are zero-average piecewise polynomials, so are the real functions corresponding to the superposition. For example, it is not difficult to show that the first integral is the familiar square wave, with amplitude $1/2$,

$$<tex2html_comment_mark>\renewedcommand{arraystretch}{2.0} ... ... 1}{\displaystyle 2}} & \mbox{for} & \theta<0, \end{array}$$

which is a piecewise linear function with two sections, having unit-height step discontinuities with opposite signs at the two singular points $\theta=0$ and $\theta=\pm\pi$.

We want to determine what is the character of the real functions, in the integration side of the resulting integral-differential chain, in the most general case, when we consider arbitrary linear superpositions of a finite number of delta ``functions'' and derivatives of delta ``functions'', with their singularities situated at various points on the unit circle. From the examples we see that, when we superpose several singular distributions with their singularities at various points, the complete set of all the singular points defines a new set of sections. Given one of these singular points, since at least one of the distributions being superposed is singular at that point, in general so is the superposition. Let there be $N\geq 1$ singular points $\{\theta_{1},\ldots,\theta_{N}\}$ in the superposition. It follows that in general we end up with a set of $N$ contiguous sections, consisting of open intervals between singular points, that can be represented as the sequence

$$\left\{ \rule{0em}{2ex} (\theta_{1},\theta_{2}), \ldots,... ...i},\theta_{i+1}), \ldots, (\theta_{N},\theta_{1}) \right\},$$ (64)

where we see that the sequence goes around the unit circle, and where we adopt the convention that each section $(\theta_{i},\theta_{i+1})$ is numbered after the singular point $\theta_{i}$ at its left end. In addition to this, since for each one of the distributions being superposed the real functions on the integration side of the integral-differential chain of the corresponding delta ``function'' are piecewise polynomials, and since the sum of any finite number of polynomials is also a polynomial, so are the real functions of the integral-differential chain to which the superposition belongs, if we are at a point in that integral-differential chain where all singular distributions have already been integrated out. Let us establish a general notation for these piecewise polynomial real functions, as well as a formal definition for them.

Definition 1   : Piecewise Polynomial Real Functions

Given a real function $f_{(n)}(\theta)$ that is defined in a piecewise fashion by polynomials in $N\geq 1$ sections of the unit circle, with the exclusion of a finite set of $N$ singular points $\theta_{i}$, with $i\in\{1,\ldots,N\}$, so that the polynomial $P_{i}^{(n_{i})}(\theta)$ at the $i^{\rm th}$ section has order $n_{i}$, we denote the function by

$$f_{(n)}(\theta) = \left\{ P_{i}^{(n_{i})}(\theta), i\in\{1,\ldots,N\} \right\},$$ (65)

where $n$ is the largest order among all the $N$ orders $n_{i}$. We say that $f_{(n)}(\theta)$ is a piecewise polynomial real function of order $n$.

Note that, being made out of finite sections of polynomials, the real function $f_{(n)}(\theta)$ is always an integrable real function. In fact, it is also analytic within each section, so that the $N$ singularities described above are the only singularities involved. Since $f_{(n)}(\theta)$ is an integrable real function, let $w(z)$ be the inner analytic function that corresponds to this integrable real function, as constructed in [#!CAoRFI!#]. The $(n+1)^{\rm th}$ angular derivative of $w(z)$ is the inner analytic function $w^{(n+1)\mbox{\Large$\cdot$}\!}(z)$, which corresponds therefore to the $(n+1)^{\rm th}$ derivative of $f_{(n)}(\theta)$ with respect to $\theta$, that we denote by $f_{(n)}^{(n+1)\prime}(\theta)$.

In this section we will prove the following theorem.

Theorem 3   : If the real function $f_{(n)}(\theta)$ is a non-zero piecewise polynomial function of order $n$, defined in $N\geq 1$ sections of the unit circle, with the exclusion of a finite non-empty set of $N$ singular points $\theta_{i}$, then and only then the derivative $f_{(n)}^{(n+1)\prime}(\theta)$ is the superposition of a non-empty set of delta ``functions'' and derivatives of delta ``functions'' on the unit circle, with the singularities located at some of the points $\theta_{i}$, and of nothing else.

Proof 3.1   :

In order to prove this, first let us note that the derivative $f_{(n)}^{(n+1)\prime}(\theta)$ is identically zero within all the open intervals defining the sections. This is so because the maximum order of all the polynomials involved is $n$, and the $(n+1)^{\rm th}$ derivative of a polynomial of order equal to or less than $n$ is identically zero,

$$f_{(n)}^{(n+1)\prime}(\theta) = 0 \mbox{\ \ \ for all\ \ \ } \theta\neq\theta_{i}, i\in\{1,\ldots,N\}.$$ (66)

We conclude, therefore, that the real object represented by the inner analytic function $w^{(n+1)\mbox{\Large$\cdot$}\!}(z)$ has support only at the $N$ isolated singular points $\theta_{i}$, thus implying that it can contain only singular distributions.

Second, let us prove that the derivative cannot be identically zero over the whole unit circle. In order to do this we note that one cannot have a non-zero piecewise polynomial real function of order $n$, such as the one described above, that is also continuous and differentiable to the order $n$ on the whole unit circle. This is so because this hypothesis would lead to an impossible integral-differential chain.

If this were possible, then starting from a non-zero real function $f_{(n)}(\theta)$ that corresponds to a non-zero inner analytic function $w(z)$, and after a finite number $n$ of steps along the differentiation direction of the corresponding integral-differential chain, one would arrive at a real function that is continuous over the whole unit circle, that is constant within each section and that has zero average over the whole unit circle. It follows that such a function would have to be identically zero, thus corresponding to the inner analytic function $w(z)\equiv 0$. But this is not possible, because this inner analytic function belongs to another chain, the one which is constant, all members being $w(z)\equiv 0$, and we have shown in [#!CAoRFI!#] that two different integral-differential chains cannot intersect.

It follows that $f_{(n)}(\theta)$ can be globally differentiable at most to order $n-1$, so that the $n^{\rm th}$ derivative is a discontinuous function, and therefore the $(n+1)^{\rm th}$ derivative already gives rise to singular distributions. Therefore, every real function that is piecewise polynomial on the unit circle, of order $n$, when differentiated $n+1$ times, so that it becomes zero within the open intervals corresponding to the existing sections, will always result in the superposition of some non-empty set of singular distributions with their singularities located at the points between two consecutive sections.

We can also establish that only functions of this form give rise to such superpositions of singular distributions and of nothing else. The necessity of the fact that the real functions on integral-differential chains generated by superpositions of singular distributions must be piecewise polynomials comes directly from the fact that all such distributions and all such superpositions of distributions are zero almost everywhere, in fact everywhere but at their singular points. Due to this, it is necessary that these real functions, upon a finite number $n+1$ of differentiations, become zero everywhere strictly within the sections, that is, within the open intervals between two successive singularities. Therefore, within each open interval the condition over the sectional real function at that interval is that

$$\frac{d^{(n+1)}}{d\theta^{(n+1)}}f_{i}(\theta) \equiv 0,$$ (67)

and the general solution of this ordinary differential equation of order $n+1$ is a polynomial of order $n_{i}\leq n$, containing at most $n+1$ non-zero arbitrary constants,

$$f_{i}(\theta) = P_{i}^{(n_{i})}(\theta).$$ (68)

Since only polynomials have the property of becoming identically zero after a finite number of differentiations, it is therefore an absolute necessity that these real functions be polynomials within each one of the sections. This completes the proof of Theorem 3.

Note that the inner analytic function $w^{(n+1)\mbox{\Large$\cdot$}\!}(z)$ corresponding to $f_{(n)}^{(n+1)\prime}(\theta)$ represents therefore the superposition of a non-empty set of singular distributions with their singularities located at the singular points. In other words, after $n+1$ angular differentiations of $w(z)$, which correspond to $n+1$ straight differentiations with respect to $\theta$ of the polynomials $P_{i}^{(n_{i})}(\theta)$ within the sections, one is left with an inner analytic function $w^{(n+1)\mbox{\Large$\cdot$}\!}(z)$ whose real part converges to zero in the $\rho\to 1_{(-)}$ limit, at all points on the unit circle which are not one of the $N$ singular points.

It is interesting to note that, since we have the inner analytic function that corresponds to the Dirac delta ``function'' in explicit form, it is not difficult to calculate directly its first angular primitive. A few simple and straightforward calculations lead to

$$w_{\delta}^{-1\mbox{\Large$\cdot$}\!}(z,z_{1}) = \frac{\mbox{\boldmath$\imath$}}{\pi} \int_{0}^{z}dz'\, \frac{z'}{z'-z_{1}}$$

$$= \frac{\mbox{\boldmath$\imath$}}{\pi} \ln\!\left(\frac{z_{1}-z}{z_{1}}\right).$$ (69)

This inner analytic function has a logarithmic singularity at $z_{1}$, which is a borderline hard singularity. Note that, as expected, we have that $w_{\delta}^{-1\mbox{\Large$\cdot$}\!}(0,z_{1})=0$.