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For the degree-two Cardinal B-spline basis on knots $1,\dots ,n+3$, sampling at $x_i=i+2$ gives the upper-bidiagonal B-spline collocation matrix $A=(I+S)/2$, where $S$ is the one-step upper shift. Since $S^n=0$, $A^{-1}=2{\sum }_{r=0}^{n-1}(-S{)}^r$ and its maximum absolute row sum is $2n$. The B-spline interpolation operator norm estimate with the order-three stability constant $d_3=3$ gives the displayed linear upper and lower bounds. The lower bound proves failure of uniform boundedness.

Let $y=(f(x_1),\dots ,f(x_n){)}^T$ and write the interpolating spline as $P_{x}f={\sum }_{j=1}^n{c}_j{N}_j$. The interpolation equations are $A_{x}c=y$, so $c=A_x^{-1}y$. Since $P_x$ is defined as a unique interpolant for arbitrary data, its B-spline collocation matrix is invertible. Ordered sites satisfying diagonal positivity ensure this by question 7. Positivity alone without admissible ordering does not: two equal sites in overlapping basis supports could give two identical rows. The norm argument below uses the existence of the operator stipulated here.

Standard partition-normalized B-splines satisfy $N_j\ge 0$ and ${\sum }_{j=1}^n{N}_j\le 1$. The latter is a subpartition of unity for B-splines: extending the knot sequence gives a full partition of unity, and discarding basis functions cannot increase its sum. HenceTaking the supremum over $t$ and then over unit-norm $f$ gives the B-spline interpolation operator norm estimateHere the matrix norm is the induced maximum-row-sum norm, not the maximum of its individual entries. The proof only needs nonnegative basis functions, their sum at most one, and invertible interpolation.

Apply the general B-spline interpolation operator norm estimate in part (I) to the inverse norm from (b):and hence in particular $\Vert P_{\mathbf{x}}{\Vert }_{L^{\infty }}\le 3$, independently of $n$. The strict finite-dimensional estimate uses the actual finite collocation matrix; it does not require periodic closure, natural boundary conditions or endpoint knot repetitions.

There is also a direct shorter verification of the uniform bound. For an index $i$ maximizing $|z_i|$, the triangle inequality giveswith an even larger margin when a neighbor is missing. This is the inverse infinity-norm bound from diagonal dominance, and again gives $\Vert A_n^{-1}{\Vert }_{{\ell }^{\infty }}\le 3$.

Use the standard partition normalization of the B-splines, so $N_j\ge 0$ and the subpartition of unity for B-splines gives ${\sum }_{j=1}^n{N}_j(t)\le 1$. Write the interpolating spline as $P_{x}f={\sum }_j{c}_j{N}_j$ and the sampled values as $y_i=f(x_i)$. The interpolation equations are $A_{x}c=y$, hence $c=A_x^{-1}y$ whenever the stated interpolation map is well defined. For every $t$,The sampling map also satisfies $\Vert y{\Vert }_{{\ell }^{\infty }}\le \Vert f{\Vert }_{\infty }$. Taking supremum norms and then the operator norm therefore provesThis 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.

Combine the B-spline interpolation operator norm estimate from part 1 with the inverse operator norm computed in part 2(b):so in particular $\Vert P_x{\Vert }_{L^{\infty }}\le 2$, uniformly in $n$. The subpartition of unity for B-splines is important at the outer boundary: the finite basis may sum to less than one there, but that only strengthens the estimate.

For the usual spline interpolation problem the sites are distinct and ordered; the Schoenberg–Whitney theorem then makes $A_x$ invertible under the stated support positivity. More generally, the following argument applies whenever the interpolation map in the question is well defined and $A_x$ is invertible.

Factor the spline interpolation operator as $P_x=B{A}_x^{-1}R$, whereSampling has operator norm at most one from the supremum norm to ${\ell }_{\infty }$. Nonnegativity and the subpartition of unity for B-splines giveConsequently $\Vert P_x\Vert \le \Vert A_x^{-1}{\Vert }_{{\ell }_{\infty }}$.

For the reverse bound, select a row $r$ of $A_x^{-1}$ with maximal absolute row sum, and choose $y_j=\mathrm{sgn}((A_x^{-1}{)}_{rj})$, assigning any value of modulus at most one when the entry is zero. Then $\Vert y{\Vert }_{{\ell }_{\infty }}=1$ andAt distinct sites there is a continuous piecewise linear function $f$ with $f(x_j)=y_j$ and $\Vert f{\Vert }_{\infty }=1$: interpolate linearly between the ordered sites and extend constantly to the endpoints. The given uniform-norm stability of a B-spline basis therefore impliesThus the B-spline interpolation operator norm satisfiesPositivity $N_i(x_i)>0$ alone, without the usual distinct-site convention, is insufficient to define an interpolation operator. For example, take order-two B-splines with knots $0,1,2,3$ and $x_1=x_2=3/2$. Both diagonal values are $1/2$, but the B-spline collocation matrix has identical rows $(1/2,1/2)$ and is singular. The displayed proof uses the well-defined interpolation map posited in the question; ordered distinct sites supply its standard existence guarantee.

For a fixed spline space and admissible interpolation sites, this linear operator sends a continuous function to its unique interpolating spline. Here $Rf$ is the vector of sampled values and $A$ is the B-spline collocation matrix. It reproduces every spline in its range. Its sensitivity to perturbing the data is quantified by the B-spline interpolation operator norm.

For a linearly independent partition-normalized B-spline basis, equivalence of norms gives a lower coefficient-stability constant. The upper bound follows from the subpartition of unity for B-splines. A knot-independent stability bound can be chosen depending on spline order. This norm comparison transfers matrix estimates to the B-spline interpolation operator norm.