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The Marsden dual functionals extract coefficients in a B-spline expansion. If ${\psi }_i$ is the knot polynomial in the Marsden identity, then on polynomials of degree at most $k-1$ one such functional isand the right-hand side is independent of $x$.

For the orthogonal B-spline projector $P$, the normal equations in the mixed-normalization spline Gram matrix give $Ga=r$, with $r_i=(M_i,f)$. Positivity and unit integral of $M_i$ imply $\Vert r{\Vert }_{{\ell }^{\infty }}\le \Vert f{\Vert }_{\infty }$. Nonnegativity and subpartition of unity of $N_i$ give $\Vert {\sum }_i{a}_i{N}_i{\Vert }_{\infty }\le \Vert a{\Vert }_{{\ell }^{\infty }}$. Hence $\Vert P{\Vert }_{\infty }\le \Vert G^{-1}{\Vert }_{{\ell }^{\infty }}$, where the matrix norm is maximum absolute row sum. The range is the spline space, which lies in $C[0,1]$ only for continuous splines.

For partition-normalized B-splines $N_i$ and integral-normalized $M_i=k{N}_i/(t_{i+k}-t_i)$, let $H_{ij}=(N_i,N_j)$ and $D_{ii}=(t_{i+k}-t_i)/k$. The mixed matrix $G_{ij}=(M_i,N_j)$ equals $D^{-1}H$. It is invertible whenever the splines form a basis and their support widths are positive. It need not be symmetric, unlike the ordinary Gram matrix.

For order $k$ B-splines, comparison of coefficients in the Marsden identity giveson the basic knot interval. Here $e_m$ is the elementary symmetric polynomial, with $e_0=1$. The $m=0$ case is partition of unity; the $m=1$ coefficients are the Greville abscissae.

A non-uniform rational B-spline surface isHere the $N_{i,p}$ and $M_{j,q}$ are B-spline basis functions for possibly nonuniform spline knot sequences, and the $w_{ij}$ are weights. Positive weights make each evaluated point a convex combination of the active control points, supplying bounding volumes for geometric searches. The denominator must remain nonzero for the parametrization to be defined.

When the knot polynomial is normalized as ${\psi }_i(x)={\prod }_{\ell =1}^{k-1}(x-t_{i+\ell })/(k-1)!$, the displayed formula has no additional factorial prefactor. Applied to Marsden's identity, it extracts the coefficient of the corresponding B-spline in any polynomial of degree at most $k-1$. Differentiating the finite sum makes adjacent terms cancel, proving independence of $x$. For linear polynomials the coefficient is their value at the Greville abscissa.

The equality holds when an alternating unit-norm spline $s_{\ast }$ has as many ordered extremal sites as the dimension, and those sites satisfy the Schoenberg–Whitney theorem. For every admissible set, sampling has operator norm at most one, giving $\kappa (\mathcal{S})\le \Vert A_{\mathbf{x}}^{-1}{\Vert }_{\infty }$. At the alternating sites the checkerboard inverse of a totally nonnegative matrix identifies each inverse absolute row sum with the magnitude of the corresponding coefficient of $s_{\ast }$. The Chebyshev spline coefficient and dual norm equality therefore gives equality. This minimizes amplification into B-spline coefficients; it does not by itself prove optimality for the distinct Lebesgue constant of interpolation.

Write $N_{i,d}$ for the partition-of-unity-normalized degree-$d$ B-spline, supported on $[t_i,t_{i+d+1}]$. Assume $d\ge 1$. First take distinct knots; endpoint repetitions will be interpreted as confluent limits. For fixed parameter $t$, let $f_t(\tau )=(\tau -t{)}_{+}^d$, and formwhere the brackets denote a divided difference in the knot variable $\tau$. This representation follows from the defining properties, rather than assuming a derivative recurrence. If $t<t_i$, $f_t$ is a degree-$d$ polynomial in all its knot arguments and its order-$(d+1)$ divided difference is zero. If $t>t_{i+d+1}$, every truncated-power term is zero. Between knots it is a degree-$d$ polynomial with $d-1$ continuous derivatives. Those support and smoothness conditions determine a one-dimensional spline space: there are $d+1$ spans, the continuity conditions leave $2d+1$ coefficients, and vanishing through order $d-1$ at both support ends imposes $2d$ conditions.

The remaining scalar is fixed by the prescribed integral. Put $a=t_i$, $b=t_{i+d+1}$. Integrating the truncated power first givesThe order-$(d+1)$ divided difference of this polynomial is its leading coefficient $1/(d+1)$, so the proposed $N_{i,d}$ has integral $(b-a)/(d+1)$, exactly the specified normalization.

Now differentiate with respect to $t$ and apply the final recursion for a divided difference:In terms of normalized lower-degree B-splines, this is the B-spline differentiation formulaA zero denominator corresponds to a collapsed-support basis term and contributes zero. Repeated knots follow by coalescing distinct knots; at knots where the classical derivative is discontinuous, read the formula on each open span or one-sided. The stated maximal interior continuity corresponds to simple interior knots, while the clamped endpoint repetitions do not affect the interior argument.

Let the control-point abscissae be the Greville abscissaeCollect the coefficient of $N_{i,d-1}$ after differentiating:The difference of the two consecutive knot averages telescopes toso every noncollapsed coefficient is one. The lower-degree basis on the endpoint-trimmed knot vector has partition of unity; equivalently the outer lower-degree basis functions on the original vector have collapsed support. Therefore $X^{\prime }(t)=1$ on the parameter interval. With $d+1$ repeated left endpoint knots at $a$, the clamped basis has $N_{0,d}(a)=1$, all other values zero, and ${\xi }_0=a$. Thus $X(a)=a$ fixes the constant of integration. Continuity and endpoint limits now giveThis is linear precision at Greville abscissae. It requires neither equal knot spacing nor a restriction on the other control-point coordinates; those determine the remaining coordinates of the parametric curve.

The cubic spline is even, so the symmetry of the normalized B-splines gives $N_{j,4}(-x)=N_{6-j,4}(x)$. It therefore suffices to extract three coefficients. The two polynomial pieces join with matching values and first two derivatives at zero, so the function belongs to the stated cubic spline space. At zero its right-hand derivatives areUse the supplied De Boor–Fix spline coefficient functional with the unnormalized knot polynomial ${\psi }_j(x)={\prod }_{r=j+1}^{j+3}(t_r-x)$. In this sign convention the coefficient isFor $j=1$, ${\psi }_1=(-1-x{)}^3$. Taking the right-hand limit at $-1$ kills the first three terms and gives $c_1=f(-1)=1$. For the next two coefficients take the right-hand limit at zero. Their knot polynomials and derivatives there areConsequently $c_2=(-27+6)/6=-7/2$ and $c_3=(27+6)/6=11/2$. Although the third derivative jumps at zero, its multiplier ${\psi }_j(0)$ vanishes for these two functionals; taking either one-sided limit gives the same coefficients. Symmetry gives $c_4=c_2$ and $c_5=c_1$. ThusFor a normalized B-spline basis, define the synthesis linear map $Tc={\sum }_j{c}_j{N}_{j,4}$ from the maximum coefficient norm into the spline space with its supremum norm. Nonnegativity and partition of unity give $\Vert T\Vert =1$, including equality on the all-ones vector. The coefficient condition number of a normalized B-spline basis isOn $[0,1]$, $f^{\prime }(x)=(27/2)x(3x-2)$. The only interior extremum is at $2/3$, where $f=-1$; at zero and one, $f=1$. Reflection therefore gives $\Vert f{\Vert }_{\infty }=1$. Its largest absolute coefficient is $11/2$, proving

Write the spline interpolation operator as $P_{\mathbf{x}}=T{A}_{\mathbf{x}}^{-1}R$, 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 $\Vert T\Vert \le 1$. HenceHere 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 giveThe three cardinal Lagrange interpolation polynomials areThe Lebesgue constant of interpolation gives the exact operator norm: the upper bound follows from $|Pg(x)|\le \Vert g{\Vert }_{\infty }{\sum }_i|{\ell }_i(x)|$. 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 $0\le x\le 1/2$, 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 $5/4$. ThusThe coefficient estimate loses cancellation among the B-splines, which explains why it is not sharp for the resulting function norm.

The Korovkin theorem says that a sequence of positive linear operators on continuous functions on $[0,1]$ converges uniformly to the identity on every continuous function if it does so on $1$, $x$ and $x^2$. Necessity is immediate; the quadratic barrier argument establishing sufficiency is given in Solution 6.

There is an endpoint misprint in the original PDF: the repeated knots at the right end must equal $1$, not $0$. Otherwise the advertised nondecreasing spline knot sequence on $[0,1]$ does not exist. Use the corrected clamped spline knot sequences, with fixed order $k$ and maximum gap $h_n\to 0$. Let $m_n$ denote the number of B-splines for the $n$th sequence. Nonnegativity and partition of unity show that the Schoenberg spline operator $V_n$ is positive and $V_{n}1=1$.

The monomial B-spline coefficients reproduce the first two nonconstant test polynomials. PutThe pair sum uses $r<s$, as in the PDF; the converted TeX incorrectly includes $r=s$. Every sampling point ${\tau }_j$ and every knot entering these coefficients lie in $[t_j,t_{j+k}]$, whose length is at most $k{h}_n$. Thus $|{\tau }_j-a_{1,j}|\le k{h}_n$. Also all these numbers belong to $[0,1]$, soThe positive B-spline weights giveTogether with $V_{n}1=1$, the three limits proveUnder the usual interior knot multiplicities at most $k-1$, the splines are continuous and the quoted form of the Korovkin theorem applies directly. If full interior multiplicity is allowed, the output can be discontinuous. The same positive-operator quadratic barrier proof still works with bounded output functions and the supremum norm, and proves exactly the stated uniform error limit without asserting continuity of each output. Indeed the stronger local-support error bound for a Schoenberg spline operator, $\Vert V_{n}g-g{\Vert }_{\infty }\le \omega (g,k{h}_n)$, follows by comparing $g(x)$ with each active sample value. The restriction $k\ge 3$ is needed for the displayed quadratic reproduction argument, not for this direct support estimate.

Fix a clamped nondecreasing spline knot sequence $t_1,\dots ,t_{n+k}$ on $[a,b]$, with $t_1=\cdots =t_k=a$ and $t_{n+1}=\cdots =t_{n+k}=b$. For continuous splines assume interior multiplicities at most $k-1$, and assume $t_i<t_{i+k}$. An order-$k$ spline has degree at most $k-1$ on each nonempty knot interval; at a knot of multiplicity $r$ it has $k-1-r$ continuous derivatives. Its dimension is $n$, and a normalized B-spline basis $N_1,\dots ,N_n$ has local support, nonnegative values and sum one on $[a,b]$.

These properties are visible in the Cox-de Boor recurrence. Starting with interval indicators at order one, setwith a term of zero denominator defined to be zero. On each contributing support the weights are nonnegative, and summing the recurrence makes the coefficients of each lower-order B-spline add to one. This gives partition of unity and $\mathrm{supp}{N}_{i,k}=[t_i,t_{i+k}]$, with endpoint values defined by limits.

At distinct ordered sites $x_1<\cdots <x_n$, interpolation is the linear systemIt has a unique solution for every data vector exactly when the B-spline collocation matrix is invertible. The Schoenberg–Whitney theorem characterizes this by $N_i(x_i)>0$ for every $i$; away from clamped endpoints this is $t_i<x_i<t_{i+k}$. At the clamped endpoints the first and last B-splines have value one, so the endpoint version uses this positive-value formulation rather than the strict inequalities. Both ordering and the support conditions matter. A spline space with interior knots is generally not a Chebyshev system on the entire interval: a locally supported B-spline vanishes on an interval. Thus arbitrary distinct nodes need not permit interpolation. Merely specifying all data sites as knots also need not fix the remaining freedom; for example, cubic interpolation with simple interior knots at $m-2$ data sites has dimension $m+2$, and two extra boundary conditions are needed to interpolate $m$ values uniquely.

The positivity governing this existence theorem is total nonnegativity of B-spline collocation matrices: every square minor with increasing row and column indices is nonnegative. In spline terminology this is often called total positivity, but it permits zero minors caused by disjoint supports. It is not the stronger assertion that every minor is strictly positive. A useful proof uses knot insertion. Each refinement step changes coefficients bywith the unchanged endpoint pieces. The refinement matrix is nonnegative bidiagonal, and its minors are nonnegative: any nonzero determinant has its allowed diagonal matching and is a product of nonnegative entries. Products preserve this property by the Cauchy–Binet formula. Insert each sampling site to full multiplicity, so evaluation at that site becomes selection of an ordered refined coefficient. The collocation matrix is then an ordered row submatrix of the refinement product, proving total nonnegativity. The strict support criterion is the strict part of this argument: an ordered path through the consecutive support intervals exists exactly when every diagonal B-spline value is positive, giving the Schoenberg–Whitney theorem determinant criterion.

If $A$ is invertible, its determinant is positive and the adjugate identity yieldsThis checkerboard inverse of a totally nonnegative matrix is the central stability fact. With alternating data $e_i=(-1{)}^i$, the absolute row sums satisfyFor general data, the cardinal splines are ${\ell }_i(x)={\sum }_j{N}_j(x)(A^{-1}{)}_{ji}$. The spline interpolation operator and its exact Lebesgue constant of interpolation areThe equality for the operator norm follows by extending the signs at distinct sites to a continuous unit-norm data function, as in Solution 2. That solution also shows the last inequality can be very strict. Stability concerns the range function as well as its coefficients.

An optimal interpolation set must be specified relative to an objective. For the normalized B-spline coefficients, every admissible sampling set satisfiessince $c(s)=A_{\mathbf{x}}^{-1}(s(x_i))$ and sampling cannot increase the supremum norm. If a unit-norm spline $s_{\ast }$ has $n$ ordered alternating extrema $s_{\ast }(x_i)=(-1{)}^i$ at admissible sites, the checkerboard identity givesTaking the largest row shows $\Vert A_{\mathbf{x}}^{-1}{\Vert }_{\infty }\le \kappa (\mathcal{S})$, so equality holds. This proves the optimal spline coefficient interpolation sites property and the Chebyshev spline coefficient and dual norm equality. In particular, for the cubic space of Solution 1 the alternating extrema $-1,-2/3,0,2/3,1$ satisfy the support conditions. Its coefficients have maximum magnitude $11/2$, so these sites are optimal for coefficient recovery andThe argument proves optimality whenever this admissible equioscillating spline is available; it does not identify extrema of an arbitrary spline as optimal sites, or transfer coefficient optimality automatically to the Lebesgue constant of interpolation.

For the intrinsic interpolation objective, minimize ${\Lambda }_{\mathbf{x}}$ itself. There is a minimizer for a fixed finite-dimensional continuous spline space. Indeed the sites lie in a compact ordered simplex. On admissible sets, inverse entries and hence ${\Lambda }_{\mathbf{x}}$ vary continuously. As a collocation matrix tends to a singular one, its inverse norm diverges. The uniform-norm stability of a B-spline basis gives the complementary boundchoose sample signs attaining a largest inverse row, extend them to a continuous unit-norm function, and use $\Vert Tc{\Vert }_{\infty }\ge \Vert c{\Vert }_{\infty }/\kappa (\mathcal{S})$. Thus a minimizing sequence with bounded Lebesgue constants of interpolation cannot approach a singular site set. A convergent subsequence has an admissible limit attaining the minimum.

Fekete interpolation sites for a continuous function space offer another useful, constructive criterion: maximize the absolute evaluation determinant. A nonzero maximum exists by compactness and linear independence. Replacing one row by evaluation at $x$ multiplies the determinant by the corresponding cardinal function, so maximality gives $|{\ell }_i(x)|\le 1$, and therefore ${\Lambda }_{\mathbf{x}}\le n$. These sites are determinant-optimal, with a proven interpolation bound; one should not claim without further argument that they minimize ${\Lambda }_{\mathbf{x}}$. Greville abscissae provide inexpensive admissible sites for the usual continuous clamped spline spaces of order at least two. Admissibility alone does not give the best stability bound.

Finally, the coefficient condition number of a normalized B-spline basis is bounded by a constant depending only on order. To see the mechanism, for each coefficient choose a longest nonempty knot cell in that B-spline's support. If its length is $h$ and the support width is $W$, then $W/h\le k$. On that cell the spline is a degree-at-most-$k-1$ polynomial; repeated Markov polynomial inequalities bound its $r$th derivative by $C_k{h}^{-r}\Vert s{\Vert }_{\infty }$. In the De Boor–Fix spline coefficient functional, the accompanying derivative of the knot polynomial is bounded by $C_k{W}^r$. Each term is consequently at most $C_k(W/h{)}^r\Vert s{\Vert }_{\infty }$, yielding $\Vert c(s){\Vert }_{\infty }\le C_k\Vert s{\Vert }_{\infty }$ independently of the number and spacing of knots. This is basis stability; it does not bound $\Vert A_{\mathbf{x}}^{-1}\Vert$ at arbitrarily poor sampling sites.

For approximation, reproduction gives the Lebesgue inequality for approximation projectorsAs the maximum knot gap tends to zero with fixed order, spline quasi-interpolation gives an approximant with error at most $\omega (f,kh)$, so the infimum tends to zero. A sequence of uniformly bounded interpolation linear operators therefore converges to every continuous target. Local support and a stable choice of sites make spline interpolation effective; mesh refinement without control of the interpolation operator norm is not by itself a convergence proof.