Basis function
==============

.. toctree::
   :maxdepth: 2
   :hidden:

   polynomial.rst
   exponential.rst
   catmullrom.rst
   cubichermite.rst
   exponentialhermite.rst


The basis function :math:`\phi` has the following important properties.

* **Support.** The *support* is the size of the largest interval in which :math:`\phi` is non-zero. For instance, the function

  .. math::
     :name: basis:eq:1

     \phi(t)=\begin{cases}
     1, \quad t\in [-\frac{1}{2}, \frac{1}{2}] \\
     0, \quad  \mathrm{elsewhere}
     \end{cases}

  has a support of size :math:`1`.
  If :math:`\phi` has a support of size :math:`L`, then :math:`\phi(t-k)` will be zero outside of :math:`[k-\frac{L}{2}, k+\frac{L}{2}]` and it will only occupy the :math:`\lceil \frac{L}{2} \rceil` intervals on each side of :math:`k`. The support thus dictates how many neighboring intervals each basis acts upon. Relying of basis that have a small support size means that each control point "controls" only a very localized portion of the entire function.

* **Interpolatory behaviour.** The basis function :math:`\phi` is said to be *interpolatory* if

  .. math::
     :name: basis:eq:2

     \phi(k) =\begin{cases}
     1, \quad k=0 \\
     0, \quad k \in \mathbb{Z}_{\ne 0}.
     \end{cases}

  When this is the case, we therefore have that :math:`r(k)=c[k]`, *i.e.* the spline and the control points coincide at integer values of :math:`k`.
  Most basis are not interpolatory and, as a consequence, spline functions constructed with them do not go through their own control points (*i.e.*, :math:`r(k) \neq c[k]`.
  Because of this, it is sometimes easier to think in terms of *knots* as opposed to control points.
  The knots :math:`n` correspond to the values of the spline at integer locations, *i.e.*

  .. math::
     :name: basis:eq:3

     n[k] = r(k).

  Therefore, if :math:`\phi` is interpolatory, the knots and the control points coincide (*i.e.*, :math:`n[k] = c[k]`).
  The knots also correspond to the junction points of the different intervals of the spline.

  Knots and control points are related through the so-called *inverse filter* of :math:`\phi`.
  Details can be found in [Unser1999]_, but for the purpose of using the \texttt{splinebox} library it is sufficient to understand that control points and knots are related, and that each of these sequences can be transformed into one another.

* **Degree and regularity.** The degree and regularity of :math:`\phi` dictates respectively the degree of the intervals and the regularity at the knots in the resulting spline.
  For instance, if :math:`\phi` is a cubic polynomial, then :math:`r` will be piecewise cubic and twice differentiable at the knots.