Hello B-Spline Family!
Illustration of the B-spline \(\beta\) and its variational forms \(\dot{\beta}\) and \(\int\beta,\) several degrees jointly.
A Family of B-Splines
B-splines are functions that are even-symmetric and blob-shaped. In the Jupyter Lab notebook below, we plot the first ten members of the family of B-splines, as ordered by their degree.
Jupyter Lab notebook
Gradient
B-splines of degree \(n\) are \(n\)-times differentiable, and continuously differentiable \(\left(n-1\right)\) times. We plot in the next Jupyter Lab the gradient (i.e., the first derivative) of the B-splines of degree \(1\) to \(9.\)
Jupyter Lab notebook
Integral
B-splines are functions that have a finite support. In their specific case, this means that these functions take the value zero for all arguments outside of some bounded interval. More precisely, \(\forall x\in{\mathbb{R}}\setminus[-\frac{n+1}{2},\frac{n+1}{2}]:\beta^{n}(x)=0\) for \(n\in{\mathbb{N}}.\) Moreover, B-splines never have any singularity. Consequently, their integral is always well-defined. We plot in the next Jupyter Lab the integral \(\int_{-\infty}^{x}\,\beta^{n}(y)\,{\mathrm{d}}y\) of the B-splines of degree \(0\) to \(9.\)
Jupyter Lab notebook