= Solution
Write the <spline interpolation operator> as $P_{\mathbf x}=TA_{\mathbf x}^{-1}R$, where $R$ samples at the interpolation sites and $T$ synthesizes the normalized <B-spline> expansion. Sampling has <operator norm> at most one, and nonnegative <B-splines> with sum at most one give $\|T\|\le1$. Hence
$$
\|P_{\mathbf x}g\|_\infty\le\|A_{\mathbf x}^{-1}Rg\|_{\ell^\infty}\le\|A_{\mathbf x}^{-1}\|_{\ell^\infty}\|g\|_\infty,
\qquad\boxed{\|P_{\mathbf x}\|\le\|A_{\mathbf x}^{-1}\|_{\ell^\infty}.}
$$
Here the <matrix norm> is the maximum absolute row sum. Existence of the <inverse matrix> requires the usual ordered distinct interpolation sites and the <Schoenberg–Whitney theorem> support conditions, with endpoint values interpreted by their one-sided limits. If the displayed diagonal condition is read without the implicit ordering and distinctness, it is insufficient: all three sites equal to $1/2$ in the quadratic example give positive diagonal entries but three identical rows. The estimate applies whenever the interpolating <linear map> in the question is defined uniquely.
For the quadratic <Bernstein basis>, $N_1=(1-x)^2$, $N_2=2x(1-x)$ and $N_3=x^2$. Sampling and inverting give
$$
A=\begin{pmatrix}1&0&0\\1/4&1/2&1/4\\0&0&1\end{pmatrix},\qquad
A^{-1}=\begin{pmatrix}1&0&0\\-1/2&2&-1/2\\0&0&1\end{pmatrix},\qquad
\boxed{\|A^{-1}\|_{\ell^\infty}=3.}
$$
The three cardinal <Lagrange interpolation polynomials> are
$$
\ell_0(x)=2x^2-3x+1,\qquad\ell_1(x)=4x(1-x),\qquad\ell_2(x)=2x^2-x.
$$
The <Lebesgue constant of interpolation> gives the exact <operator norm>: the upper bound follows from $|Pg(x)|\le\|g\|_\infty\sum_i|\ell_i(x)|$. To attain it at a maximizing point, prescribe at the three sites the signs of the corresponding cardinal <polynomials> and extend those values by a <continuous> <piecewise linear function> of <norm> one.
On $0\le x\le1/2$, the first two cardinal <polynomials> are nonnegative and the last is nonpositive. Since their sum is one, their absolute sum is $1-2\ell_2(x)=1+2x-4x^2$. Its maximum is $5/4$ at $x=1/4$. Reflection gives the same maximum at $3/4$ on the other half. In particular the data $(1,1,-1)$ produce $1+2x-4x^2$, attaining $5/4$. Thus
$$
\boxed{\|P_{\mathbf x}\|=\frac54<3=\|A^{-1}\|_{\ell^\infty}.}
$$
The coefficient estimate loses cancellation among the <B-splines>, which explains why it is not sharp for the resulting <function> <norm>.
Back to article page
