to symmetry of B-splines on uniform meshes, the collocation matrices are also symmetric and Töplitz. So, we can conclude that cardinal B-spline collocation matrix is Töplitz, symmetric and positive definite.

It is easy to show that the elements of the collocation matrix do not depend on the step-size $h$ of the uniform mesh, so we take the simplest one with $h = 1$ and $x_i = i$. Such B-splines are called cardinal. The interpolation nodes (\[2.3\]) are integers for odd degrees, while for even degrees, the interpolation nodes are in the middle of the two neighbouring knots of the cardinal B-spline.

![The first four cardinal splines $N_i^d$ of degree $d = 3$ and $4$, respectively. Big black dots denote the spline values at the knot averages.[]{data-label="fig2.1"}](BS3B7.eps "fig:"){width="5.75cm"} ![The first four cardinal splines $N_i^d$ of degree $d = 3$ and $4$, respectively. Big black dots denote the spline values at the knot averages.[]{data-label="fig2.1"}](BS4B8.eps "fig:"){width="5.75cm"}

The normalized basic cardinal spline $N^d$ suffices to determine all basis function values at the interpolation nodes $$N_i^d (x_j^{\ast}) = N^d(x_{j-i}^{\ast}).$$

The general de Boor–Cox reccurence relation [@deBoor-2001], written in terms of the degree of a spline is: $$N^d(x) = \frac{x N^{d - 1}(x) + (d + 