= Marsden identity
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= Marsden's identity
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{synonym}
For partition-normalized order-$k$ <B-splines> on the basic knot interval, <Marsden's identity> states
$$
(y-x)^{k-1}=\sum_i\psi_{i,k}(y)N_{i,k}(x),\qquad\psi_{i,k}(y)=\prod_{r=1}^{k-1}(y-t_{i+r}).
$$
It 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>.<Math CS 887: Approximation Theory by Carl de Boor>[Marsden identity (93)–(94), printed p.57; equivalent simultaneous sign convention]{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 $\sum_iN_{i,2}(x)=1$ and $\sum_it_{i+1}N_{i,2}(x)=x$. Thus the identity encodes exact reproduction of low-degree <polynomials> and leads to the <Marsden dual functional>.
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