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This linear functional extracts one coefficient of a B-spline expansion using local polynomial data. With ${\psi }_i(x)={\prod }_{\ell =1}^{k-1}(x-t_{i+\ell })/(k-1)!$ and a point $\xi$ inside a knot cell in the support, it is ${\sum }_{r=0}^{k-1}(-1{)}^r{s}^{(r)}(\xi ){\psi }_i^{(k-1-r)}(\xi )$. The local polynomial identity for the Marsden dual functional gives the desired coefficient. Its bounded restriction to the local spline space can be extended to continuous data on $[t_i,t_{i+k}]$ by the Hahn-Banach theorem, with the same bound. Such extensions supply spline quasi-interpolation coefficients without requiring derivatives of the data.

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.

Put $m=k-1$, so the Marsden identity isDifferentiating $m-j$ times with respect to $x$ givesThe exact Taylor formula for a polynomial isSubstitution of the differentiated Marsden identities yieldswhere

Differentiating this expression for ${\lambda }_i$ produces two sums whose adjacent terms cancel. The uncancelled endpoints contain ${\psi }_i^{(k)}$ and $p^{(k)}$, both zero because the two functions have degree at most $k-1$. Thusso the Marsden dual functional ${\lambda }_i(p)$ is independent of the auxiliary point $x$.