# Higher-level functions

The functions listed below help with common use cases of B-splines. See their docstrings for more information.

### knotaverages/knotaverages!

The knotaverages function returns a vector $\tau$ of knot averages

$$$\tau_i = \frac{1}{k-1} \sum_{j=i+1}^{i+k-1} t_j$$$

where $t$ is the knot vector of the B-spline basis and $k$ is its order. The length of $\tau$ equals the number of B-splines in the basis. The knot averages are recommended in [deBoor1978] (p. 214) as data points for interpolation. Instead of creating the knot vector for all indices $i$, a range of indices can be supplied with the keyword argument indices.

The knotaverages! function can be used to write the knot averages to a pre-allocated array.

### averagebasis

The averagebasis function returns a BSplineBasis of a specified order that is well-suited for interpolating a function at a given set of data points.

Warning

To obtain a valid B-spline basis, the data points must be sorted in ascending order. This is not checked by the averagebasis function.

### basismatrix/basismatrix!

The basismatrix function calculates the matrix

$$$B_{ij} = B_j(x_i)$$$

where $B_j$ is the $j$-th B-spline of a basis and $x$ is a vector of real numbers. The range of indices $j$ can be supplied with the indices keyword argument.

The basismatrix! function can be used to write $B_{ij}$ to a pre-allocated array.

### interpolate

The interpolate function interpolates data (vectors of $x$ and $y$ values) in a given B-spline basis. It returns a Spline. The indices keyword can be used to restrict the interpolation to a range of B-splines from the basis.

### approximate

The approximate function approximates a function $f:\mathbb{R}\to\mathbb{R}$ in a given B-spline basis by sampling the function at the knot averages of the basis and interpolating the samples. The indices keyword can be used to restrict the B-splines used for the interpolation.

• deBoor1978Carl de Boor, A Practical Guide to Splines, New York, N.Y.: Springer, 1978.