= Solution
For unit-spaced knots, the <quadratic cardinal B-spline> with support $[i,i+3]$ is $N_i(t)=B(t-i)$, where the <Cox-de Boor recurrence> gives
$$
B(u)=\begin{cases}u^2/2,&0\le u\le1,\\-u^2+3u-3/2,&1\le u\le2,\\(3-u)^2/2,&2\le u\le3,\\0,&\text{otherwise}.\end{cases}
$$
In particular $B(1)=B(2)=1/2$ and $B(0)=B(3)=0$. At $x_i=i+2$, the only nonzero entries in row $i$ are $A_{ii}=1/2$ and, if $i<n$, $A_{i,i+1}=1/2$. Write $U$ for the upper shift matrix, with ones immediately above its diagonal. Then
$$
A_x=\tfrac12(I+U),\qquad A_x^{-1}=2\sum_{r=0}^{n-1}(-U)^r,
$$
since $U^n=0$. Thus $(A_x^{-1})_{ij}=2(-1)^{j-i}$ for $j\ge i$, and zero otherwise. The largest absolute row sum is the first, giving $\|A_x^{-1}\|_{\ell_\infty}=2n$.
Using (1) and the permitted $d_3=3$ proves the stronger <linear growth of shifted quadratic spline interpolation> estimate
$$
\boxed{\frac{2n}{3}\le\|P_x\|\le2n.}
$$
Therefore \b[$\|P_x\|=\Theta(n)$ and is not uniformly bounded]. An $O(n)$ upper bound alone would not imply unboundedness; the lower bound is essential.
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