Signs, Zeros, and Zero-Crossings
How to access the signed pieces of a piecewise-polynomial periodic one-dimensional spline and how to take advantage of them.
Terminology
Signum Function
The signum function is the complex-to-complex mapping
It maps the origin of the complex plane to itself, and it maps every other point to its closest point on the unit circle. Those points \(z\in{\mathbb{C}}\) such that \({\mathrm{sgn}}(z)=1\) are said to be positive, and those points \(z\in{\mathbb{C}}\) such that \({\mathrm{sgn}}(z)=\left(-1\right)\) are said to be negative; the set of the real numbers can then be partitioned as \({\mathbb{R}}=\{z\in{\mathbb{C}}:{\mathrm{sgn}}(z)=-1\}\cup\{0\}\cup\{z\in{\mathbb{C}}:{\mathrm{sgn}}(z)=1\}.\)
Zeros
Consider the piecewise polynomial spline \(f:{\mathbb{R}}\rightarrow{\mathbb{R}}\) of nonnegative integer degree \(n.\) If an interval \({\mathbb{X}}\) can be found such that it is of maximal diameter and such that \(\forall x\in{\mathbb{X}}:f(x)=0,\) then this interval is said to be a zero of the spline.
Zero-Crossings
Spline of Null Degree
Consider a spline of degree \(n=0.\) If a proper interval \({\mathbb{X}}\) can be found such that \(\forall x\in{\mathbb{X}}:f(x)=0\) and such that \(f(\inf{\mathbb{X}})\,f(\sup{\mathbb{X}})<0,\) then the interval is said to be a zero-crossing of the spline.
If two proper intervals \({\mathbb{A}}\) and \({\mathbb{B}}\) can be found such that they are mutually disjoint, such that \(\sup{\mathbb{A}}=\inf{\mathbb{B}},\) and such that \(\forall a\in{\mathbb{A}},\forall b\in{\mathbb{B}}:f(a)\,f(b)<0,\) then, the degenerate interval \({\mathbb{X}}=\{x\in{\mathbb{R}}:\sup{\mathbb{A}}=x=\inf{\mathbb{B}}\}\) is said to be a zero-crossing of the spline.
Spline of Positive Degree
Consider a spline of degree \(n>0.\) If three intervals \({\mathbb{A}},\) \({\mathbb{X}},\) and \({\mathbb{B}}\) can be found such that they are mutually disjoint, such that \(\left(\sup{\mathbb{A}}=\inf{\mathbb{X}}\right)\wedge\left(\sup{\mathbb{X}}=\inf{\mathbb{B}}\right),\) such that \(\forall x\in{\mathbb{X}}:{\mathrm{sgn}}(f(x))=0,\) and such that \(\forall a\in{\mathbb{A}},\forall b\in{\mathbb{B}}:f(a)\,f(b)<0,\) then, \({\mathbb{X}}\) is said to be a zero-crossing of the spline.
Descending or Ascending Zero-Crossings
A descending zero-crossing is such that \(\lim_{x\uparrow\inf{\mathbb{X}}}{\mathrm{sgn}}(f(x))>\lim_{x\downarrow\sup{\mathbb{X}}}{\mathrm{sgn}}(f(x)).\)
An ascending zero-crossing is a zero-crossing that is not descending.
Spline Signed Pieces
Given a periodic spline of period \(K,\) we want to establish a set of intervals that partitions the main period \([0,K)\) in such a way that each element of the set is an interval of largest diameter over which the spline takes a constant-valued sign. For ease of use, the set is returned as a list.
We now propose a few lines of code that create and display a random spline and extract its constant-sign pieces.
Jupyter Lab notebook
Spline Zeros and Zero-Crossings
Given a periodic spline, we want to establish a set of intervals of largest diameter that give either its zeros or its zero-crossings. We enforce that the diameter of the enclosure of all returned intervals is not larger than the period of this spline. For ease of use, the set is returned as a list.
We now propose a few lines of code that create and display a curated spline over which we illustrate and discuss several configurations, represented by a mixture of degenerate and proper intervals. In our control of the configurations, we retain the same spline coefficients but we let the degree of the spline vary.
Jupyter Lab notebook
In the next notebook, we examine random splines of specified period, degree, and delay. Here, it is the values of the spline samples that we maintain across configurations.
Jupyter Lab notebook