= Solution
Use the standard partition normalization of the <B-splines>, so $N_j\ge0$ and the <subpartition of unity for B-splines> gives $\sum_{j=1}^nN_j(t)\le1$. Write the interpolating <spline> as $P_xf=\sum_jc_jN_j$ and the sampled values as $y_i=f(x_i)$. The interpolation equations are $A_xc=y$, hence $c=A_x^{-1}y$ whenever the stated interpolation map is well defined. For every $t$,
$$
|P_xf(t)|\le\sum_j|c_j|N_j(t)
\le\|c\|_{\ell^\infty}\sum_jN_j(t)
\le\|c\|_{\ell^\infty}.
$$
The sampling map also satisfies $\|y\|_{\ell^\infty}\le\|f\|_\infty$. Taking <supremum norms> and then the <operator norm> therefore proves
$$
\boxed{\|P_x\|_{L^\infty\to L^\infty}\le\|A_x^{-1}\|_{\ell^\infty\to\ell^\infty}.}
$$
This is the <B-spline interpolation operator norm> bound, and it applies on any interval containing the sampling sites on which these normalized <B-splines> are used.
There is a small qualification to the printed hypotheses: diagonal positivity alone does not ensure uniqueness unless the usual site assumptions are supplied. For example, with two quadratic <B-splines> on knots $1,2,3,4,5$ and $x_1=x_2=3$, both diagonal values equal $1/2$, but $A_x$ has two identical rows and is singular. The intended increasing sites $x_1<\cdots<x_n$ with $N_i(x_i)>0$ satisfy the <Schoenberg–Whitney theorem>, making $A_x$ invertible; alternatively invertibility is implicit in the premise that $P_x$ is the unique interpolation map. The displayed bound is proved under that premise, rather than inferring invertibility from diagonal positivity alone.
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