Spline approximation

A spline quasi-interpolant synthesizes a B-spline expansion from local bounded linear functionals, instead of solving a global interpolation system. Suppose each functional is bounded by $c_k\Vert f{\Vert }_{C[t_i,t_{i+k}]}$ and the operator reproduces every spline. On a knot cell $[t_j,t_{j+1}]$, only indices $j+1-k\le i\le j$ contribute. Their supports lie in $[t_{j+1-k},t_{j+k}]$, and subpartition of unity for B-splines proves local operator norm at most $c_k$. Reproduction of degree-$k-1$ polynomials and a local Taylor theorem remainder then give error at most $(1+c_k)(2k-1{)}^k{h}^k\Vert f^{(k)}{\Vert }_{\infty }/k!$, on the basic knot domain with the usual completed boundary basis.

For a spanning direction matrix $\Xi =({\xi }_1,\dots ,{\xi }_m)$ in ${\mathbb{R}}^d$, the box spline is the density of the pushforward of uniform measure on $[0,1{]}^m$ under $t\mapsto \Xi t$. Equivalently,Its support is the zonotope ${\sum }_j[0,1]{\xi }_j$. It is piecewise polynomial of total degree at most $m-d$. For integer directions, splitting each integration interval in half yields its binary refinement symbolup to the monomial specifying the choice of origin. If $r$ is the smallest number of directions whose removal leaves a nonspanning set, the usual box spline smoothness criterion gives $C^{r-2}$ regularity. Repeated directions increase smoothness while enlarging support.

The unit-direction univariate box spline is the density of the sum of $m$ independent uniform unit-interval coordinates. It is the order-$m$ Cardinal B-spline: its closed support is $[0,m]$, it is strictly positive on the interior, and its polynomial degree is $m-1$. For $m\ge 2$ it is $C^{m-2}$, generally not $C^{m-1}$. The recurrence $B_m(x)={\int }_0^1{B}_{m-1}(x-u)du$ proves its support and positivity. Starting with a half-open unit interval, integrating the translate partition recursively proves ${\sum }_{j\in \mathbb{Z}}{B}_m(x-j)=1$. Orthogonal projection onto the unit diagonal rescales support width from $m$ to $\sqrt{m}$.

This box spline is the density of the image of the uniform unit four-cube under the indicated direction matrix. Its polynomial pieces have total degree $4-2=2$. Removing three directions is necessary to leave a nonspanning set, giving $C^1$ continuity. Its knot arrangement is a criss-cross triangulation of a square lattice. Its refinement mask is $\frac{1}{4}(1+z)(1+w)(1+zw)(1+z/w)$, up to the monomial that fixes the origin.

Take two copies of each direction $(1,0),(0,1),(1,1),(1,-1)$. The resulting box spline has total polynomial degree six and $C^4$ continuity: there are eight directions and a rank-deficient remaining set has at most two, so the removal number is six. With a centred origin its support is the zonotopeIts centred binary symbol isThis symbol factors as $A(z,w)A(zw,z/w)$ with $A=(2+z+z^{-1})(2+w+w^{-1})/8$, giving an efficient quincunx subdivision implementation. The box spline criterion is also described in the primary research paper 4–8 Subdivision by Velho and Zorin.

A quadratic spline with simple knots is a continuously differentiable function that is a polynomial of degree at most two between successive knots. Its first derivative is continuous and piecewise linear, while its second derivative is piecewise constant and can jump at knots. Constant tails impose vanishing first derivatives at the outermost knots. Such a spline arises from first-derivative roughness penalization when observations are interval averages, as in the quadratic smoothing spline for interval averages.

Spline interpolation matches prescribed function values at distinct sites by an element of a finite-dimensional spline space. In a B-spline basis its existence and uniqueness for all data are equivalent to invertibility of the B-spline collocation matrix. Distinct ordered sites satisfying the Schoenberg–Whitney theorem give this condition.

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 unit-spaced order-three B-splines sampled at support midpoints $i+3/2$, the B-spline collocation matrix has diagonal $3/4$ and neighboring entries $1/8$. Conjugating by alternating diagonal signs reduces its inverse absolute values to the nonnegative inverse of $\mathrm{tridiag}(-1,6,-1)/8$. With $\tau =3-2\sqrt{2}$, the absolute row sums are $r_i=2[1-({\tau }^i+{\tau }^{n+1-i})/(1+{\tau }^{n+1})]$. This follows by solving $6r_i-r_{i-1}-r_{i+1}=8$ with $r_0=r_{n+1}=0$. The maximum occurs at the middle row or rows and tends to two as $n$ grows. Nonnegativity and subpartition of unity of the B-splines transfer this bound to the spline interpolation operator.

For distinct interpolation sites and an invertible B-spline collocation matrix, the interpolant is $P=B{A}^{-1}R$, where $R$ samples data and $Ba={\sum }_i{a}_i{N}_i$. Sampling and basis synthesis have operator norms at most one in the supremum norm. For the reverse inequality, realize the signs of a maximal absolute row sum of $A^{-1}$ as values of a continuous function of norm one, and apply uniform-norm stability of a B-spline basis. This yields both displayed bounds.

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.

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.

A regression spline fits a regression function in a finite-dimensional spline approximation space with selected knots. Evaluating its basis at predictor values gives a design matrix, reducing fitting to regression on basis coefficients.

A cubic regression spline represents a mean function in a finite-dimensional space of piecewise cubic functions, with continuity of the function and its first two derivatives at chosen knots. A penalized fit minimizes a residual criterion plus $\lambda \int (f^{\prime \prime }{)}^2$, using ${\Omega }_{jk}=\int B_j^{\prime \prime }{B}_k^{\prime \prime }$. Unlike a full cubic smoothing spline with knots at every distinct observation, its knot set can be much smaller. The mgcv basis bs="cr" uses a penalized natural cubic regression-spline basis with linear tails.

A penalized regression spline estimates basis coefficients by minimizing a residual or likelihood criterion plus a curvature penalty. Basis size limits the available complexity; the penalty controls how much of that complexity is used.

The nonnegative functional $J(g)={\int }_a^b(g^{\prime \prime }(x){)}^{2}dx$. Its null space on $C^2[a,b]$ consists of affine functions. The natural cubic spline interpolant uniquely minimizes this penalty among functions matching at least two prescribed knot values. It is a squared second derivative penalty, not the exact geometric curvature energy of a graph.

A junction between adjacent polynomial pieces of a cubic spline or other spline. Continuity conditions at these junctions determine which piecewise polynomial functions are splines.

A spline knot sequence is a nondecreasing list of spline knots, including their multiplicities, that specifies the junctions and smoothness constraints of a spline space. An order-$k$ B-spline uses $k+1$ consecutive knot entries. Its support must have positive width, and repeated knots are interpreted by confluent divided differences or their limiting convention.

A cubic spline is a twice continuously differentiable piecewise polynomial of degree at most three whose polynomial pieces meet at specified knots.

A natural cubic spline is linear beyond its two outer knots, equivalently its second derivative vanishes at those knots.

For at least two distinct knots $x_1<\cdots <x_n$ and values $v_i$, the unique natural cubic spline matching those values and linear beyond $x_1,x_n$. Its interpolation map is a linear map by uniqueness. Over a larger domain $[a,b]$, the two exterior pieces remain linear. With only one knot, arbitrary slopes of affine functions make uniqueness false.

Let $b_i=N{e}_i$ be the interpolating cardinal basis and set ${\Gamma }_{ij}={\int }_a^b{b}_i^{\prime \prime }{b}_j^{\prime \prime }dx$. Linearity gives $J(N\mathbf{v})={\mathbf{v}}^T\Gamma \mathbf{v}$, and this Gram matrix is a positive semidefinite matrix. Its null space consists precisely of vectors $(\alpha +\beta x_i{)}_i$, by the null-space statement for the second derivative roughness penalty. Thus its rank is $n-2$, including the zero matrix when $n=2$.

For $g=N\mathbf{v}$ and any $\tilde{g}\in C^2[a,b]$ taking the same values at at least two distinct knots, $J(\tilde{g})=J(g)+J(\tilde{g}-g)$. To prove it, set $r=\tilde{g}-g$, integrate $g^{\prime \prime }{r}^{\prime \prime }$ by parts on each polynomial interval and use $g^{(4)}=0$, $r(x_i)=0$, continuous $g^{\prime \prime }$ and linear exterior pieces. Boundary terms cancel. Expanding the square proves the identity. Equality forces $r^{\prime \prime }=0$, so $r$ is an affine function and its two prescribed zeros make it identically zero.

The minimum roughness property of the natural cubic spline interpolant extends to interpolants with an absolutely continuous first derivative and square-integrable second derivative. For $h=f-g$ vanishing at the knots, integration by parts on each cubic interval gives $\int g^{\prime \prime }{h}^{\prime \prime }=0$: the knot boundary terms cancel by continuity, the exterior terms vanish by natural boundary conditions, and the remaining constant-third-derivative term vanishes because $h$ is zero at the interval endpoints. Expanding the square proves minimality. Equality makes $h$ affine; its zeros at two distinct knots force $h=0$, proving uniqueness. Competitors with infinite roughness cannot improve the finite spline penalty.

A B-spline is a compactly supported piecewise polynomial basis function determined by consecutive knots.

For the partition-of-unity normalization, the derivative of a degree-$d$ B-spline is the displayed difference of two overlapping degree-$(d-1)$ B-splines. A zero denominator denotes a term with collapsed support, assigned zero. For simple knots the divided-difference representation $N_{i,d}(t)=(t_{i+d+1}-t_i)[t_i,\dots ,t_{i+d+1}](\tau -t{)}_{+}^d$ has the required support, smoothness and integral normalization. Differentiate the truncated power and apply the last-step divided difference recursion to obtain the formula. Repeated-knot versions follow by confluent limits, with one-sided interpretation where necessary.

The $k$ order-$k$ B-splines active on a simple-knot cell form a basis of its degree-$k-1$ polynomials. To see this from Marsden's identity, use the dual polynomials ${\psi }_i(y)={\prod }_{\ell =1}^{k-1}(y-t_{i+\ell })$. On $t_j<x<t_{j+1}$ the identity is $(y-x{)}^{k-1}={\sum }_{i=j-k+1}^j{N}_i(x){\psi }_i(y)$. Evaluation of the dual polynomials at $t_j,t_{j-1},\dots ,t_{j-k+1}$ gives a triangular coefficient matrix with nonzero diagonal. They are independent, and comparison of coefficients of $y$ in the identity gives independence of the active splines. Consequently a spline vanishes on a whole open cell exactly when all its active coefficients vanish.

Let the real spline $s={\sum }_{i=p}^q{c}_i{N}_i$ have distinct knots, order $k\ge 2$, nonzero endpoint coefficients, and no identically zero knot cell in its support. It and its first $k-2$ derivatives vanish at both support endpoints. If $Z$ is its number of interior distinct zeros, there are initially $Z+2$ zero components. Each differentiation through order $k-2$ increases this count by at least one: a nonzero gap between zero components has an interior extremum, and the two endpoint zeros remain separate. The final derivative is continuous piecewise linear on $q-p+k$ cells, and each cell can meet at most one zero component unless it is identically zero, in which case all its zeros belong to the same component. It therefore has at most $q-p+k$ zero components. Thus $Z+k\le q-p+k$, proving the bound.

For a uniform quadratic B-spline with interior control points $P_i$, label its $i$th span $[i,i+1]$. A quartic B-spline with interior spline knot sequence ${\tau }_{3i}={\tau }_{3i+1}={\tau }_{3i+2}=i$ represents the identical curve whenTo verify this, raise each quadratic Bézier curve span twice by degree elevation of Bernstein coefficients. The three interior quartic controls are the displayed points. One additional knot insertion at each interior knot supplies the shared endpoint as the mean of its two neighboring controls, recovering all five quartic Bézier curve controls. Thus the knot multiplicity is three, giving $C^1$ joins rather than imposing unwanted extra smoothness.

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.

Knot insertion refines the spline knot sequence and changes the control points without changing the represented B-spline. Repeated insertion exposes local Bézier curves and provides subdivision-based evaluation.

// Target: analysis.bigb

For degree $p$, knots $t_i$ and span $[t_k,t_{k+1})$, initialize $d_j^{(0)}=P_{k-p+j}$ for $0\le j\le p$. At level $r$, update $j=p,p-1,\dots ,r$ by $d_j^{(r)}=(1-{\alpha }_{j,r})d_{j-1}^{(r-1)}+{\alpha }_{j,r}{d}_j^{(r-1)}$, where ${\alpha }_{j,r}=(t-t_{k-p+j})/(t_{k+1+j-r}-t_{k-p+j})$. The point is $d_p^{(p)}$. This evaluates a B-spline by local affine combinations.

// Target: analysis.bigb

For strictly increasing spline knots, $M_i(t)=k[t_i,\dots ,t_{i+k}](\cdot -t{)}_{+}^{k-1}$ is a nonnegative integral-normalized B-spline. Integrating the truncated power function over its support replaces its spline knot value by $(u-t_i{)}^k/k$. The order-$k$ divided difference of this degree-$k$ polynomial is its leading coefficient $1/k$. Multiplying by the prefactor $k$ in $M_i$ gives integral one. The corresponding partition-normalized B-spline is $N_i=(t_{i+k}-t_i)M_i/k$.

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.

For a nonnegative partition-normalized B-spline basis, the synthesis linear map $Tc={\sum }_j{c}_j{N}_j$ has operator norm one. Its condition number is consequently $\Vert T^{-1}\Vert$, with the supremum norm on the spline space and the maximum norm on its coefficients. The uniform-norm stability of a B-spline basis bounds this number independently of the spline knot sequence when the order is fixed. This is different from the condition number of a particular sampled B-spline collocation matrix.

Suppose $\Vert s_{\ast }{\Vert }_{\infty }=1$ and $s_{\ast }(x_j)=(-1{)}^j$ at increasing sites satisfying the Schoenberg–Whitney theorem conditions. For the B-spline collocation matrix $A$, the coefficient linear functional in the dual basis is ${\mu }_i(s)={\sum }_j(A^{-1}{)}_{ij}s(x_j)$. Therefore $\Vert {\mu }_i\Vert \le {\sum }_j|(A^{-1}{)}_{ij}|$. The checkerboard inverse of a totally nonnegative matrix makes ${\mu }_i(s_{\ast })=(-1{)}^i{\sum }_j|(A^{-1}{)}_{ij}|$, attaining that bound. Since ${\mu }_i(s_{\ast })=a_i^{\ast }$,The conclusion is an extremal property of an alternating unit-norm spline, not a claim that every bounded spline has these extremal coefficients.

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.

A finite collection of the standard partition-normalized B-splines is nonnegative and has sum at most one on the entire real line. Extend the knot sequence in both directions. The full order-one collection sums to one, and summing the Cox-de Boor recurrence preserves that sum because the two coefficients of each lower-order function add to one. The finite collection is a subset. On the basic knot interval the finite sum equals one, also following from Marsden's identity.

For distinct increasing knots and $k\ge 2$, the explicit divided difference formula writes an order-$k$ B-spline as a finite sum of truncated power functions of degree $k-1$. It is a piecewise polynomial function and is globally $C^{k-2}$. Below its first knot the order-$k$ divided difference annihilates a degree-$k-1$ polynomial; above its last knot all values vanish. Nonzero first and last pieces give the exact closed support. Order one gives interval indicators instead of continuous splines.

An order-$k$ B-spline is supported on $[t_i,t_{i+k}]$. Below this interval its truncated-power knot data agree with a degree-$k-1$ polynomial, annihilated by the order-$k$ divided difference; above it all data vanish. Therefore its ordinary or mixed Gram matrix has entries zero when $|i-j|\ge k$. The half-bandwidth is $k-1$, and for strictly increasing knots this bound is sharp.

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.

Let nonnegative partition-normalized B-splines $N_i$ form a basis, with ${\sum }_i{N}_i\le 1$ on the integration interval and $d_i=\int N_i>0$. The row-normalized mixed Gram matrix is $G_{ij}=\int N_i{N}_j/d_i$. The normal equations for the orthogonal projection onto their linear span become $Ga=b$, with $b_i=\int f{N}_i/d_i$ and $\Vert b{\Vert }_{{\ell }^{\infty }}\le \Vert f{\Vert }_{\infty }$. The subpartition of unity for B-splines gives $\Vert {\sum }_i{a}_i{N}_i{\Vert }_{\infty }\le \Vert a{\Vert }_{{\ell }^{\infty }}$ and proves the bound. The ordinary symmetric Gram matrix omits this essential row normalization. The projection maps continuous functions into the spline space, not onto all continuous functions.

For the distinct-knot linear B-spline hats, let $h_i=t_{i+1}-t_i>0$. The mixed-normalization spline Gram matrix has entriesOnly indices in the finite basis are retained. This matrix has a row strict diagonal dominance margin at least $1/3$, so the inverse infinity-norm bound from diagonal dominance gives $\Vert G^{-1}{\Vert }_{{\ell }^{\infty }}\le 3$. Equidistant spline knots give off-diagonal entries $1/6$. Both pieces of every hat are retained; repeated endpoint spline knots require different boundary formulas.

For distinct increasing knots, let ${\ell }_i(\cdot ,t)$ interpolate $(\cdot -t{)}_{+}^{k-1}$ on $t_i,\dots ,t_{i+k-1}$ and let ${\omega }_i(x)={\prod }_{r=1}^{k-1}(x-t_{i+r})$. Adjacent interpolants share $k-1$ values; their difference is a multiple of ${\omega }_i$. Its leading coefficient, by the divided difference recurrence, is the order-$k$ B-spline $N_i(t)$. Thus ${\ell }_{i+1}(x,t)-{\ell }_i(x,t)={\omega }_i(x)N_i(t)$. Telescoping proves the Marsden identity. Repeated knots require the well-defined confluent or limiting convention.

A cardinal B-spline is a B-spline whose knots are consecutive integers.

Sampling consecutive order-six Cardinal B-splines at their support midpoints gives the displayed finite B-spline collocation matrix. No exterior columns are retained. Its row strict diagonal dominance margin is at least $(66-2\cdot 26-2)/120=1/10$, so the inverse infinity-norm bound from diagonal dominance gives $\Vert A^{-1}{\Vert }_{{\ell }^{\infty }}\le 10$. The uniform knot samples are $N_{0,6}(0),\dots ,N_{0,6}(6)=(0,1,26,66,26,1,0)/120$, obtained recursively from the Cox-de Boor recurrence.

This degree-two Cardinal B-spline has three consecutive unit knot intervals as its support. Its first and last polynomial pieces are $(t-i{)}^2/2$ and $(i+3-t{)}^2/2$; on the middle interval it equals $3/4-(t-i-3/2{)}^2$. Its two interior integer values are $1/2$, and its endpoint values are zero. The maximum is $3/4$, emphasizing that partition normalization does not mean unit maximum height.

Sampling order-three Cardinal B-splines $N_j$ at $x_i=i+3/2$ gives $N_i(x_i)=3/4$, $N_{i-1}(x_i)=N_{i+1}(x_i)=1/8$, and all other entries zero. The B-spline collocation matrix therefore has row strict diagonal dominance margin at least $1/2$, giving inverse operator norm at most two. More precisely, with $q=3-2\sqrt{2}$, its absolute inverse row sums are $v_i=2[1-(q^i+q^{n+1-i})/(1+q^{n+1})]$. To see this, change signs by $D_{ii}=(-1{)}^i$: $DAD$ has negative off-diagonal entries and a nonnegative inverse given by a convergent Neumann series. Its inverse row sums solve $6v_i-v_{i-1}-v_{i+1}=8$, $v_0=v_{n+1}=0$. Thus $\Vert A^{-1}{\Vert }_{\infty }=v_{\lfloor (n+1)/2\rfloor }<2$, tending to two as the dimension grows.

A cardinal cubic B-spline is a degree-three Cardinal B-spline. At the three interior integer knots, its nonzero normalized values are $1/6$, $4/6$, and $1/6$.

For the unit-spaced cubic cardinal B-spline on $[0,4]$, the four scalar cubic Bernstein basis coefficient rows are $(0,0,0,1/6)$, $(1/6,1/3,2/3,2/3)$, $(2/3,2/3,1/3,1/6)$ and $(1/6,0,0,0)$. Each row applies on its interval with local parameter in $[0,1]$. They result from the compactly supported $C^2$ truncated-cube combination $\frac{1}{6}{\sum }_{j=0}^4(-1{)}^j(\genfrac{}{}{0ex}{}{4}{j})(x-j{)}_{+}^3$. Each graph piece is a cubic Bézier curve with abscissa controls $j,j+1/3,j+2/3,j+1$.

Sampling consecutive Cardinal cubic B-splines at the centers of their supports gives diagonal values $2/3$ and adjacent values $1/6$, with all other values zero. Finite truncation gives the displayed B-spline collocation matrix, including its endpoint rows without wrapping or adding boundary conditions. Its inverse has alternating signs. The exact inverse norm of midpoint cubic spline collocation is less than three for each finite dimension and approaches three as the dimension grows.

Conjugate the collocation matrix by the diagonal matrix with entries $(-1{)}^i$. The resulting negative-off-diagonal matrix has a nonnegative inverse, as a convergent Neumann series shows. The absolute inverse row sums are therefore the entries of its inverse applied to the all-ones vector. They solve $4u_i-u_{i-1}-u_{i+1}=6$ with $u_0=u_{n+1}=0$, whose solution is displayed. The maximum occurs at the central row or central two rows and equals the inverse operator norm on ${\ell }^{\infty }$. Consequently the associated spline interpolation operator has norm at most three independently of dimension.

The Cox-de Boor formula recursively expresses an order-$k$ B-spline using two adjacent order-$(k-1)$ B-splines.

The Marsden identity expands $(x-t{)}^{k-1}$ in a B-spline basis with knot-polynomial coefficients.

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.

For an order-$k$ B-spline with $k\ge 2$, its Greville abscissa is the arithmetic mean of its $k-1$ interior knots. These are precisely the coefficients reproducing the linear function in the monomial B-spline coefficients formula. The irregular plural is Greville abscissae.

For an open clamped spline knot sequence and degree $d\ge 1$, the Greville abscissae ${\xi }_i=d^{-1}{\sum }_{r=1}^d{t}_{i+r}$ reproduce the parameter. The B-spline differentiation formula gives derivative coefficients $d({\xi }_i-{\xi }_{i-1})/(t_{i+d}-t_i)=1$. Their lower-degree partition of unity makes the derivative of the weighted sum equal to one. The left endpoint value is the left parameter endpoint, fixing the integration constant to zero. This proves linear precision even with unequal knot intervals.

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 $x$.

This linear functional extracts one coefficient of a B-spline expansion using local polynomial data. With ${\psi }_i(x)={\prod }_{\ell =1}^{k-1}(x-t_{i+\ell })/(k-1)!$ and a point $\xi$ inside a knot cell in the support, it is ${\sum }_{r=0}^{k-1}(-1{)}^r{s}^{(r)}(\xi ){\psi }_i^{(k-1-r)}(\xi )$. The local polynomial identity for the Marsden dual functional gives the desired coefficient. Its bounded restriction to the local spline space can be extended to continuous data on $[t_i,t_{i+k}]$ by the Hahn-Banach theorem, with the same bound. Such extensions supply spline quasi-interpolation coefficients without requiring derivatives of the data.

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.

For B-splines $N_j$ and interpolation sites $x_i$, the B-spline collocation matrix is $A_{ij}=N_j(x_i)$. Its invertibility is the existence-and-uniqueness condition for interpolation in that B-spline basis.

For increasing sites and an ordered nonnegative normalized B-spline basis, the matrix $A_{ij}=N_j(x_i)$ is a totally nonnegative matrix. One way to see the minors' signs is knot insertion. Each inserted knot changes coefficients by $b_j={\alpha }_j{a}_j+(1-{\alpha }_j)a_{j-1}$ with $0\le {\alpha }_j\le 1$, using the standard constant endpoint pieces. This is a nonnegative rectangular bidiagonal map, whose minors are nonnegative. Products preserve that property by the Cauchy–Binet formula. Insert each site to full knot multiplicity; a suitable refined coefficient is then the value $s(x_i)$. Thus collocation is an ordered row submatrix of the refinement map, and has nonnegative minors. The strict support conditions $t_i<x_i<t_{i+k}$ give invertibility by the Schoenberg–Whitney theorem; its inverse has the checkerboard inverse of a totally nonnegative matrix sign pattern.

For increasing interpolation sites $x_i$ and an order-$k$ B-spline basis with knots $t_i$, the B-spline collocation matrix is invertible exactly whenfor every $i$, equivalently when $N_i(x_i)>0$ for every $i$.

A Schoenberg spline operator is a positive quasi-interpolant ${\sum }_{i}f({\tau }_i)N_i$ with each sample point inside the corresponding B-spline support.

For nonnegative partition-normalized order-$k$ B-splines, let $Vf(t)={\sum }_{i}f({\tau }_i)N_i(t)$ with each sample point inside $[t_i,t_{i+k}]$. If $N_i(t)$ is nonzero, both $t$ and ${\tau }_i$ lie in an interval of length at most $k|\Delta |$. Their function values differ by at most the modulus of continuity at that length. Summing with nonnegative weights of total one proves the displayed bound. With fixed order and vanishing maximum knot gap, uniform continuity gives uniform convergence. The same argument works for bounded piecewise-continuous outputs if full interior knot multiplicities are allowed; their individual continuity should not then be asserted.