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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 $x$.

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 $x$.

Let $e_m$ denote the $m$th elementary symmetric polynomial of $t_{i+1},\dots ,t_{i+k-1}$. Expanding both sides of the Marsden identity in powers of $x$ and equating the coefficient of $x^{k-1-m}$ givesThus $a_{0,i}=1$, while $a_{1,i}$ and $a_{2,i}$ are respectively the means of the interior knots and of their pairwise products.