= De Boor–Fix spline coefficient functional
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{title2=$\lambda_i(s)=c_i\quad\text{for }s=\sum_jc_jN_j$}
= De Boor–Fix functional
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{synonym}
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 $\sum_{r=0}^{k-1}(-1)^rs^{(r)}(\xi)\psi_i^{(k-1-r)}(\xi)$. 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.
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