Marsden dual functional

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 $x$.

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.

When the knot polynomial is normalized as ${\psi }_i(x)={\prod }_{\ell =1}^{k-1}(x-t_{i+\ell })/(k-1)!$, the displayed formula has no additional factorial prefactor. Applied to Marsden's identity, it extracts the coefficient of the corresponding B-spline in any polynomial of degree at most $k-1$. Differentiating the finite sum makes adjacent terms cancel, proving independence of $x$. For linear polynomials the coefficient is their value at the Greville abscissa.