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A family of $n+1$ real continuous functions is a Chebyshev system exactly when every evaluation matrix at $n+1$ distinct points is invertible. A singular evaluation matrix supplies a nonzero coefficient vector whose function vanishes at those points; conversely, a violation of the zero bound supplies such a vector in the kernel. This also gives the existence and uniqueness of interpolation from its span.

Exponentials with distinct real exponents form a Chebyshev system on every nondegenerate real interval. For an inductive proof, multiply a candidate combination by $e^{-{\lambda }_{0}x}$. If it has $n+1$ zeros, Rolle's theorem gives at least $n$ zeros of its derivative, a combination of the $n$ exponentials with exponents ${\lambda }_i-{\lambda }_0$, $i>0$. The inductive zero bound forces all those coefficients to vanish. The remaining constant also vanishes, a contradiction.

A Chebyshev system of real continuous functions on a circle has odd dimension. If its dimension is $d$, take $d$ cyclically ordered evaluation points and move them continuously to the next cyclic permutation without letting points collide. The evaluation determinant stays nonzero, so cannot change sign. The final cyclic row permutation multiplies it by $(-1{)}^{d-1}$; consequently $d$ must be odd. For example, $1,\cos x,\sin x,\dots ,\cos mx,\sin mx$ has dimension $2m+1$ and the required zero bound.

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.

Assume $a<b$. A Chebyshev system of dimension $n+1$ is a family of real continuous functions $u_0,\dots ,u_n$ for which every linear combination $u={\sum }_{j=0}^n{c}_j{u}_j$ with coefficients not all zero has at most $n$ distinct zeros in the interval. In particular the functions are linearly independent: an identically zero nontrivial combination would violate that bound. Multiplicities are not counted in this definition.

For distinct points $x_0,\dots ,x_n$, form the evaluation matrix $A=(u_j(x_i){)}_{i,j=0}^n$. If its determinant vanished, a nonzero vector $c$ would satisfy $Ac=0$. Its associated combination would vanish at all $n+1$ points, contradicting the Chebyshev system property. Conversely, if the property failed, some nonzero coefficient vector would give a combination vanishing at $n+1$ distinct points. At those points $Ac=0$, so the evaluation matrix would be singular. We have proved the determinant criterion for a Chebyshev system:This is also the existence-and-uniqueness condition for interpolation from the span of the functions: any prescribed $n+1$ data values give a uniquely solvable linear system.

Let $d$ be the dimension of a proposed real Chebyshev system on the circle, and regard its functions as continuous $2\pi$-periodic functions on the real line. Choose lifted points ${\theta }_0<{\theta }_1<\cdots <{\theta }_{d-1}<{\theta }_0+2\pi$ and set ${\theta }_d={\theta }_0+2\pi$. Move the points byFor every $s$ the points remain distinct on the circle. Indeed, each consecutive gap is a convex combination of positive gaps; the closing gap is also a convex combination of the original closing gap and ${\theta }_1-{\theta }_0$. The evaluation determinantis therefore nonzero for every $s$ by the determinant criterion for a Chebyshev system. It is continuous, so the intermediate value theorem forbids a sign change. At $s=1$, however, periodicity makes its rows the cyclic permutation of the initial rows, giving $D(1)=(-1{)}^{d-1}D(0)$. For even $d$ this reverses the sign, a contradiction. Hence every such periodic Chebyshev system has odd dimension; none has even dimension. This determinant argument avoids assuming differentiability or simple zeros of the functions.