Piecewise Polynomial Real Functions

In [#!CAoRFII!#] we introduced the concept of piecewise polynomial real functions, in the context of the discussion of singular Schwartz distributions. These are the integrable real functions that are obtained on the integration side of the integral-differential chains to which the singular distributions belong, the latter being on the differentiation side of the same chains, with respect to the central position of the delta ``function'' itself. This concept will return to the discussion in this paper, where it will play an essential role. Therefore, let us review the definitions and the relevant results established in that previous paper.

Let us assume that we have a real function which is defined by polynomials in sections of the unit circle. These sections are open intervals separated by a finite set of points which are singularities of that real function. Let there be $N\geq 1$ singular points $\{\theta_{1},\ldots,\theta_{N}\}$. It follows that in this case we must separate the unit circle into a set of $N$ contiguous sections, consisting of open intervals between the singular points, that can be represented as the sequence

$$\left\{ \rule{0em}{2ex} (\theta_{1},\theta_{2}), \ldots,... ...i},\theta_{i+1}), \ldots, (\theta_{N},\theta_{1}) \right\},$$ (5)

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 [#!CAoRFII!#] we established a general notation for these piecewise polynomial real functions, as well as a formal definition for them. Here is the definition: 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 non-empty 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

$$f_{(n)}(\theta) = \left\{ P_{i}^{(n_{i})}(\theta), i\in\{1,\ldots,N\} \right\},$$ (6)

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.

These functions have some important properties, which were established in [#!CAoRFII!#]. First and foremost, if the integrable real function $f_{(n)}(\theta)$ is a non-zero piecewise polynomial function of order $n$, then and only then its derivative $f_{(n)}^{(n+1)\prime}(\theta)$ with respect to $\theta$ is a superposition of a non-empty set of delta ``functions'' and derivatives of delta ``functions'' on the unit circle, with their singularities located at some of the points $\theta_{i}$, and of nothing else. In particular, the derivative $f_{(n)}^{(n+1)\prime}(\theta)$ is identically zero within all the open intervals defining the sections. However, this derivative cannot be identically zero over the whole unit circle. In fact, it is impossible to 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. It follows that $f_{(n)}(\theta)$ can be globally differentiable at most to order $n-1$, so that its $n^{\rm th}$ derivative is a discontinuous function, and therefore its $(n+1)^{\rm th}$ derivative already gives rise to singular distributions.

In short, 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. Furthermore, only functions of this form give rise to such superpositions of singular distributions and of nothing else.