[proofplan]
The upper estimate follows from the non-negativity and partition-of-unity property of the full normalized B-spline basis. For the lower estimate, we invoke the precise de Boor-Fix coefficient-functional theorem: for fixed degree $r$, the B-spline coordinate functionals are uniformly bounded on the spline space with a bound depending only on $r$. Applying the coordinate functional corresponding to a maximal coefficient recovers that coefficient from the spline and gives the lower estimate. The constants are therefore independent of all knot locations and spacings.
[/proofplan]
[step:Use positivity and partition of unity to prove the upper bound]
Let $\mathcal I$ denote the full finite index set of normalized degree $r$ B-splines associated with the given ordered extended knot vector on $[a,b]$. Thus each $N_{i,r}: [a,b] \to [0,\infty)$ is the normalized degree $r$ B-spline with index $i$. For any index omitted from the displayed expansion, set $c_i := 0$, and define $M := \max_{i \in \mathcal I}|c_i|$. The normalized B-splines satisfy $N_{i,r}(x) \ge 0$ for every $i \in \mathcal I$ and every $x \in [a,b]$, and the full family forms a [partition of unity](/page/Partition%20of%20Unity) on $[a,b]$:
\begin{align*}
\sum_{i \in \mathcal I} N_{i,r}(x)=1.
\end{align*}
Therefore, for each $x \in [a,b]$, the triangle inequality and non-negativity give
\begin{align*}
|s(x)| \le \sum_{i \in \mathcal I}|c_i|N_{i,r}(x) \le M\sum_{i \in \mathcal I}N_{i,r}(x)=M.
\end{align*}
Taking the supremum over $x \in [a,b]$ gives
\begin{align*}
\|s\|_\infty \le M.
\end{align*}
Thus the upper estimate holds with $B_r := 1$.
[guided]
The upper bound is the part of the theorem where normalization is used directly, so the partition of unity must be taken over the full B-spline family for the knot vector. Let $\mathcal I$ be the full finite index set of normalized degree $r$ B-splines on $[a,b]$. If a basis function is not written in the original finite expansion, we assign its coefficient the value $0$, so the same spline is written as $s = \sum_{i \in \mathcal I} c_i N_{i,r}$. Set $M := \max_{i \in \mathcal I}|c_i|$.
Since the B-splines are normalized, they satisfy two structural facts: each $N_{i,r}: [a,b] \to [0,\infty)$ is non-negative, and the full family indexed by $\mathcal I$ sums to one at every point of the interval:
\begin{align*}
\sum_{i \in \mathcal I} N_{i,r}(x)=1.
\end{align*}
For a fixed point $x \in [a,b]$, apply the triangle inequality to the finite sum defining $s(x)$:
\begin{align*}
|s(x)| \le \sum_{i \in \mathcal I}|c_i|N_{i,r}(x).
\end{align*}
Because $|c_i| \le M$ for every $i \in \mathcal I$ and $N_{i,r}(x) \ge 0$, each summand is bounded by $M N_{i,r}(x)$. Hence
\begin{align*}
|s(x)| \le M\sum_{i \in \mathcal I}N_{i,r}(x)=M.
\end{align*}
This estimate is uniform in $x$, so taking the supremum over $[a,b]$ yields $\|s\|_\infty \le M$. Therefore the right-hand inequality holds with $B_r := 1$, and this constant depends only on $r$ because it is independent of every parameter.
[/guided]
[/step]
[step:Invoke the de Boor-Fix coefficient-functional theorem with its exact hypotheses]
We use the following standard form of the de Boor-Fix theorem, proved from the total positivity of B-spline collocation matrices in de Boor and Fix, "Spline approximation by quasiinterpolants", Journal of [Approximation Theory](/page/Approximation%20Theory) 8 (1973), 19-45, DOI: https://doi.org/10.1016/0021-9045(73)90080-9. For each integer $r \ge 0$ there exists a constant $C_r \ge 1$ with the following property. If $T$ is an ordered extended knot vector on an interval $[a,b]$ with no multiplicity exceeding $r+1$, if $\{N_{i,r}: [a,b]\to[0,\infty)\}_{i\in\mathcal I}$ is the full normalized degree $r$ B-spline basis associated with $T$, and if $\mathcal S_r$ is the real [vector space](/page/Vector%20Space) spanned by this basis with norm $\|v\|_\infty := \sup_{x \in [a,b]}|v(x)|$, then for every $i\in\mathcal I$ there is a linear coefficient functional $\lambda_i: \mathcal S_r\to\mathbb R$ satisfying
\begin{align*}
\lambda_i(N_{j,r}) = \delta_{ij}
\end{align*}
where $\delta_{ij}$ denotes the Kronecker delta, so $\delta_{ij}=1$ when $i=j$ and $\delta_{ij}=0$ when $i\ne j$.
and
\begin{align*}
|\lambda_i(v)|\le C_r\|v\|_\infty
\end{align*}
for every $v\in\mathcal S_r$. The constant $C_r$ depends only on $r$.
In the present theorem, the knot vector is ordered, the multiplicity bound is at most $r+1$, and the B-splines are the full normalized degree $r$ family on $[a,b]$. Hence all hypotheses of the de Boor-Fix coefficient-functional theorem are satisfied, so the functionals $\lambda_i$ exist with the stated uniform bound.
[/step]
[step:Apply the dual functional at a maximal coefficient]
Choose an index $i_0 \in \mathcal I$ such that $|c_{i_0}| = M$. The spline $s$ belongs to $\mathcal S_r$, so the dual functional estimate gives a [linear map](/page/Linear%20Map) $\lambda_{i_0}: \mathcal S_r \to \mathbb R$ with $|\lambda_{i_0}(v)| \le C_r\|v\|_\infty$ for every $v \in \mathcal S_r$ and with $\lambda_{i_0}(N_{j,r}) = 1$ if $j=i_0$ and $\lambda_{i_0}(N_{j,r}) = 0$ otherwise. By linearity and finiteness of the expansion,
\begin{align*}
\lambda_{i_0}(s)=\sum_{j \in \mathcal I} c_j\lambda_{i_0}(N_{j,r})=c_{i_0}.
\end{align*}
Therefore
\begin{align*}
M=|c_{i_0}|=|\lambda_{i_0}(s)| \le C_r\|s\|_\infty.
\end{align*}
Thus
\begin{align*}
C_r^{-1}M \le \|s\|_\infty.
\end{align*}
[guided]
The lower bound is the substantive part of the theorem: we must recover the largest coefficient from the function $s$ itself with a constant that does not depend on the knot spacings. Let $i_0 \in \mathcal I$ be chosen so that $|c_{i_0}| = M$, where $M = \max_{i \in \mathcal I}|c_i|$.
The de Boor-Fix coefficient-functional theorem applies to the full spline space $\mathcal S_r$ for the given knot vector. Its hypotheses have been verified in the preceding step: the knot vector is ordered, the multiplicities are at most $r+1$, and the splines are the full normalized degree $r$ B-spline basis. Hence there is a linear map $\lambda_{i_0}: \mathcal S_r \to \mathbb R$ such that $|\lambda_{i_0}(v)| \le C_r\|v\|_\infty$ for every $v \in \mathcal S_r$ and such that $\lambda_{i_0}$ extracts the $i_0$ coefficient on the B-spline basis.
We now apply this functional to the actual spline $s$, not to a truncated local piece. Since the expansion is finite and $s = \sum_{j \in \mathcal I} c_jN_{j,r}$, linearity gives
\begin{align*}
\lambda_{i_0}(s)=\sum_{j \in \mathcal I} c_j\lambda_{i_0}(N_{j,r})=c_{i_0}.
\end{align*}
The norm bound for $\lambda_{i_0}$ then yields
\begin{align*}
M=|c_{i_0}|=|\lambda_{i_0}(s)| \le C_r\|s\|_\infty.
\end{align*}
Rearranging gives
\begin{align*}
C_r^{-1}M \le \|s\|_\infty.
\end{align*}
This avoids any unsupported restriction to a support block: the coefficient is extracted from the full spline, so no outside B-spline contributions are lost.
[/guided]
[/step]
[step:Choose constants and conclude the global stability estimate]
Define
\begin{align*}
A_r := C_r^{-1}.
\end{align*}
Define
\begin{align*}
B_r := 1.
\end{align*}
The preceding steps prove
\begin{align*}
A_r \max_{i \in \mathcal I}|c_i| \le \|s\|_\infty \le B_r \max_{i \in \mathcal I}|c_i|.
\end{align*}
Because omitted coefficients were defined to be zero, $\max_{i \in \mathcal I}|c_i|$ is the same as the maximum over the coefficients appearing in the original expansion. The constant $B_r$ is absolute, and $A_r$ depends only on the degree $r$ through the de Boor dual-functional constant $C_r$. Since $C_r$ is uniform over all ordered knot vectors with multiplicities at most $r+1$, both constants are independent of the knot locations and spacings. This proves the theorem.
[/step]
