B-spline

A B-spline is a compactly supported piecewise polynomial basis function determined by consecutive knots.

A cardinal B-spline is a B-spline whose knots are consecutive integers.

A cardinal cubic B-spline is a degree-three Cardinal B-spline. At the three interior integer knots, its nonzero normalized values are $1/6$, $4/6$, and $1/6$.

The Cox-de Boor formula recursively expresses an order-$k$ B-spline using two adjacent order-$(k-1)$ B-splines.

The Marsden identity expands $(x-t{)}^{k-1}$ in a B-spline basis with knot-polynomial coefficients.

The Marsden dual functionals extract coefficients in a B-spline expansion. If ${\psi }_i$ is the knot polynomial in the Marsden identity, then on polynomials of degree at most $k-1$ one such functional isand the right-hand side is independent of $x$.

For B-splines $N_j$ and interpolation sites $x_i$, the B-spline collocation matrix is $A_{ij}=N_j(x_i)$. Its invertibility is the existence-and-uniqueness condition for interpolation in that B-spline basis.

For increasing interpolation sites $x_i$ and an order-$k$ B-spline basis with knots $t_i$, the B-spline collocation matrix is invertible exactly whenfor every $i$, equivalently when $N_i(x_i)>0$ for every $i$.

A Schoenberg spline operator is a positive quasi-interpolant ${\sum }_{i}f({\tau }_i)N_i$ with each sample point inside the corresponding B-spline support.