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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.

Carl de Boor authored notes on approximation theory developing best approximation, B-splines and spline interpolation. The notes connect concrete recurrence formulas with structural criteria such as the Schoenberg–Whitney theorem. This makes it possible to analyze when a spline representation exists and is unique, rather than relying only on numerical fitting.^[ref]

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.

Let $K$ be a compact Hausdorff space, $U$ a real linear subspace of $C(K,\mathbb{R})$, $f\in C(K,\mathbb{R})$ and $u\in U$. Put $e=f-u$ and $E=\{x:|e(x)|=\Vert e{\Vert }_{\infty }\}$. Then $u$ is a best uniform approximation from $U$ precisely when every $v\in U$ has some $x\in E$ with $e(x)v(x)\le 0$.^[ref]

If a direction $v$ has strictly matching error signs everywhere on $E$, compactness makes $u+tv$ a uniformly better approximant for sufficiently small $t>0$. Conversely, for any candidate $u+v$, an active point with $e(x)v(x)\le 0$ has $|e(x)-v(x)|\ge |e(x)|$, preventing improvement. The linear subspace hypothesis ensures that differences of candidates are admissible directions; for a general nonlinear family, the quantified direction set would have to change. This criterion yields the Chebyshev alternation theorem.

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.

The Peano kernel theorem converts polynomial exactness into a derivative-based error formula. Let $r\ge 1$ and let $L$ be a linear error functional annihilating every polynomial of degree below $r$. A useful sufficient setting is a finite combination of point values, derivatives of orders below $r$, and integrals against finite weights. For $f\in C^r[a,b]$,Here $(x-t{)}_{+}^{r-1}$ means $(x-t{)}^{r-1}$ for $x>t$ and zero for $x<t$; values at its finitely many evaluation breakpoints can be chosen arbitrarily for the outer integral. The proof applies $L$ to Taylor's theorem with integral remainder: the polynomial part vanishes, and the permitted evaluations and integrations interchange with the remainder integral. For a more general linear functional, that interchange is an additional hypothesis. The same construction makes B-splines Peano kernels of divided differences.^[ref]

For the midpoint error $L(f)={\int }_a^{b}f-(b-a)f((a+b)/2)$, use $r=2$. Its nonnegative kernel equals $(t-a{)}^2/2$ on the left half and $(b-t{)}^2/2$ on the right half. Its integral is $(b-a{)}^3/24$, immediately yielding the sharp quadrature error bound.

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 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.