In some course it's said that a B-spline curve can be made to interpolate through one of its control points by increasing the multiplicity of a knot to n+1 (n being the degree of the curve). This can be made clear using the basic properties of splines, such as the locality property.
It is subsequently remarked that a multiplicity of n already suffices. I couldn't find any explanation as to why this is true. Does anyone here have one?
Finally found an answer. Let $u_i, ..., u_{i+n-1}$ be equal (multiplicity of $n$), then in $u_i$ the curve equals :
\begin{align} \vec{s}(u_i) & = \sum_{j=-n}^{p-1}\vec{d}_jN_j^n(u_i) \\ & = \sum_{j=i-n}^i\vec{d}_jN_j^n(u_i) \qquad\text{(locality)}\\ & = \sum_{j=i-n}^{i-2}\vec{d}_jN_j^n(u_{i+n-1})+\vec{d}_{i-1}N_{i-1}^n(u_i)+\vec{d}_iN_i^n(u_i) \qquad(u_i=u_{i+n-1})\\ \end{align}
The first term can be omitted due to the locality property, the last one because $N_i^n(u_i)$ equals zero if $u_i$ has a multiplicity lower than $n+1$ (which is given). What remains is the second term, where $N_{i-1}^n$ has to be 1 because of the summation-to-one property.