 Marsden identity

For partition-normalized order-$k$ B-splines on the basic knot interval, Marsden's identity statesIt holds for every real $y$, with the empty product equal to one at $k=1$. The variable $x$ lies in the interval where the completed B-spline basis has partition of unity.^[ref]

Comparing polynomial coefficients in $y$ yields the monomial B-spline coefficients. At $k=2$, the equation $y-x={\sum }_i(y-t_{i+1})N_{i,2}(x)$ gives both ${\sum }_i{N}_{i,2}(x)=1$ and ${\sum }_i{t}_{i+1}{N}_{i,2}(x)=x$. Thus the identity encodes exact reproduction of low-degree polynomials and leads to the Marsden dual functional.