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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 is
$$
\lambda_i(p)=\sum_{j=0}^{k-1}(-1)^j\psi_i^{(k-1-j)}(x)p^{(j)}(x),
$$
and the right-hand side is independent of $x$.

Marsden dual functional

Write the interpolating spline as
$$
P_{\mathbf x}f=\sum_{j=1}^nc_jN_j.
$$
The interpolation equations are $A_{\mathbf x}c=f_{\mathbf x}$, where $(f_{\mathbf x})_i=f(x_i)$, so
$$
\|c\|_{\ell_\infty}
\leq\|A_{\mathbf x}^{-1}\|_{\ell_\infty}
\|f_{\mathbf x}\|_{\ell_\infty}
\leq\|A_{\mathbf x}^{-1}\|_{\ell_\infty}\|f\|_\infty.
$$
The <B-splines> are nonnegative and form a <partition of unity> on the spline interval. Consequently
$$
|P_{\mathbf x}f(t)|
\leq\sum_j|c_j|N_j(t)
\leq\|c\|_{\ell_\infty}.
$$
Taking the supremum over $t$ and then over $\|f\|_\infty\leq1$ proves the <operator norm> bound
$$
\boxed{\|P_{\mathbf x}\|_{L_\infty}
\leq\|A_{\mathbf x}^{-1}\|_{\ell_\infty}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 3 / b / Solution

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A B-spline is a compactly supported piecewise polynomial basis function determined by consecutive knots.


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