Spline interpolation

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.