Bases [module]

Here are the data of the class splinekit.bases.

class splinekit.bases.Bases

Bases: object

The class that codifies the standard bases of splines.

The uniform splines are piecewise-polynomial functions of some nonnegative degree \(n.\) They can be written as the weighted sum of the integer shifts of some basis function. In linear algebra, several systems of coordinates can span the same space and the actual values of the coordinates of a fixed vector depend on the actual system of coordinates. Likewise, the expression of a fixed spline \(f\) depends on the actual basis. With common bases, the same spline can be expressed indifferently as

\[\begin{split}\begin{eqnarray*} f(x)&=&\sum_{k\in{\mathbb{Z}}}\,c[k]\,\beta^{n}(x-k)\\ &=&\sum_{k\in{\mathbb{Z}}}\,f(k)\,\eta^{n}(x-k)\\ &=&\sum_{k\in{\mathbb{Z}}}\,g[k]\,\mathring{\beta}^{n,n}(x-k)\\ &=&\sum_{k\in{\mathbb{Z}}}\,a[k]\,\phi^{n}(x-k). \end{eqnarray*}\end{split}\]

In general, while the spline \(f\) remains the same, the coefficients are mutually different, with \(f\neq c\neq g\neq a\wedge c\neq a\neq f\neq g.\)

For splinekit.Bases.BASIC, the coefficients are \(c\) and the basis is the polynomial B-spline \(\beta^{n},\) so that

\[f(x)=\sum_{k\in{\mathbb{Z}}}\,c[k]\,\beta^{n}(x-k).\]
For splinekit.Bases.CARDINAL, the coefficients are \(f\) and the basis is the cardinal B-spline \(\eta^{n},\) so that

\[f(x)=\sum_{k\in{\mathbb{Z}}}\,f(k)\,\eta^{n}(x-k).\]
For splinekit.Bases.DUAL, the coefficients are \(g\) and the basis is the polynomial dual B-spline \(\mathring{\beta}^{n,n},\) so that

\[f(x)=\sum_{k\in{\mathbb{Z}}}\,g[k]\,\mathring{\beta}^{n,n}(x-k).\]
For splinekit.Bases.ORTHONORMAL, the coefficients are \(a\) and the basis is the polynomial orthonormal B-spline \(\phi^{n},\) so that

\[f(x)=\sum_{k\in{\mathbb{Z}}}\,a[k]\,\phi^{n}(x-k).\]