B-spline ($N_{i,k}$)

An order-$k$ B-spline $N_{i,k}$ is a local, nonnegative spline determined by $k+1$ consecutive entries $t_i,\dots ,t_{i+k}$ of a nondecreasing spline knot sequence. Its polynomial pieces have degree at most $k-1$, and its support is contained in $[t_i,t_{i+k}]$. The Cox-de Boor recurrence gives a constructive definition from interval indicators.^[ref]

For example, the order-two B-spline with spline knots $0,1,2$ is the triangular hatOnly nearby B-splines are nonzero at a given point. This local support makes spline interpolation matrices and spline Gram matrices sparse. On the basic knot interval, a completed B-spline basis has partition of unity; spline knot multiplicities determine the smoothness across joins.

A de Boor point is a control point in a B-spline representation. For a degree-two uniform segment with three active de Boor points $D_0,D_1,D_2$, the equivalent Bézier curve handles are $(D_0+D_1)/2,D_1,(D_1+D_2)/2$. The next segment shares $D_1,D_2$ and introduces one new handle. This formula assumes simple equally spaced spline knots; clamped or repeated spline knots have different endpoint conversions.

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.

This degree-three Cardinal B-spline has support $[0,4]$, is twice continuously differentiable, and has values $0,1/6,2/3,1/6,0$ at the consecutive integer knots. Their sum is one. Its truncated power function representation has coefficients whose first four moments vanish, making it zero after its final knot. Four knot intervals are the minimum possible compact support for a nonzero cubic $C^2$ spline: with fewer intervals the four moment constraints form a full-column-rank Vandermonde matrix. Translating and rescaling this function gives a local basis for cubic spline interpolation on a uniform mesh.

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 centered Cardinal cubic B-spline $\beta$ with knot values $1/6,4/6,1/6$, write $s(x)={\sum }_{j=-1}^{m+1}{c}_j\beta (x/h-j)$ on $[0,mh]$. interpolation gives $c_{i-1}+4c_i+c_{i+1}=6u(ih)$ for $i=0,\dots ,m$. Since ${\beta }^{\prime \prime }(0)=-2$ and ${\beta }^{\prime \prime }(\pm 1)=1$, the natural endpoint conditions $s^{\prime \prime }(0)=s^{\prime \prime }(mh)=0$ give the displayed extra two rows. These supply the two equations missing from interpolation alone.

For knot spacing $h=1/2$ on $[0,1]$, the five translates indexed $-3,-2,-1,0,1$ span the $C^2$ piecewise cubic space. With normalized knot values $1/6,2/3,1/6$, nodal interpolation gives $(c_{-3}+4c_{-2}+c_{-1})/6=u(0)$ and the two shifted equations at $h,2h$. Left-end Hermite data give $(c_{-1}-c_{-3})/(2h)=u^{\prime }(0)$ and $(c_{-3}-2c_{-2}+c_{-1})/h^2=u^{\prime \prime }(0)$. These five equations uniquely determine the coefficients. Derivative data need existence or separate prescription; continuity of $u$ alone does not supply them.

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.

For a nondecreasing spline knot sequence $(t_i)$, initialize the order-one B-splines by $N_{i,1}(x)=1_{[t_i,t_{i+1})}(x)$. For $k\ge 2$, the Cox-de Boor recurrence isA term with zero denominator is assigned zero. For clamped endpoint evaluation, take the appropriate one-sided continuous value rather than applying the half-open order-one convention beyond the final knot.^[ref]

Each order-$k$ B-spline is assembled from two neighboring order-$(k-1)$ B-splines. On their active spans the weights are nonnegative, which proves nonnegativity inductively and keeps the support local. For simple equally spaced spline knots and $k=2$, this produces the triangular hat. Evaluating a whole spline at one point can combine these same local affine steps through De Boor's algorithm.

For partition-normalized order-$k$ B-splines on the basic knot interval, Marsden's identity statesIt holds for every real $y$, with the empty product equal to one at $k=1$. The variable $x$ lies in the interval where the completed B-spline basis has partition of unity.^[ref]

Comparing polynomial coefficients in $y$ yields the monomial B-spline coefficients. At $k=2$, the equation $y-x={\sum }_i(y-t_{i+1})N_{i,2}(x)$ gives both ${\sum }_i{N}_{i,2}(x)=1$ and ${\sum }_i{t}_{i+1}{N}_{i,2}(x)=x$. Thus the identity encodes exact reproduction of low-degree polynomials and leads to the Marsden dual functional.

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 strictly increasing spline knots $t_1<\cdots <t_{n+k}$ and interpolation sites $x_1<\cdots <x_n$, the order-$k$ B-spline collocation matrix $A_{ij}=N_{j,k}(x_i)$, with $k\ge 2$, is invertible exactly whenThese conditions place each site in the open support of its corresponding B-spline, and are equivalent to $N_{i,k}(x_i)>0$ in this simple-knot setting. They characterize existence and uniqueness of spline interpolation for all data.^[ref]

With repeated clamped endpoint spline knots, endpoint conventions matter: strict inequalities at the outer endpoints are not necessary. For example, the order-two clamped sequence $(0,0,1,1)$ gives the basis $1-x,x$ on $[0,1]$. Sampling at $0,1$ gives the identity matrix, even though the corresponding outer strict inequalities fail. One must use the version allowing nonzero endpoint B-spline values instead of importing the simple-knot inequalities unchanged.

For an order-$k$ partition-normalized B-spline basis with $k\ge 2$, the standard Schoenberg spline operator samples at the Greville abscissae:It is a positive linear operator on continuous functions on the basic knot interval. Partition of unity gives $S1=1$, and linear precision at Greville abscissae gives $Sx=x$. These reproduction properties rely on the specified samples; choosing arbitrary points in the individual supports does not generally reproduce a linear function.^[ref]

Local support gives a direct approximation bound. If every active B-spline at $x$ has $|{\xi }_i-x|\le h$, then positivity and partition of unity give $|Sf(x)-f(x)|\le \omega (f,h)$, where $\omega$ is the modulus of continuity. Hence mesh refinement with shrinking active support widths yields uniform convergence for every continuous function, while avoiding a global interpolation solve.

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.