Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 61 / 5

Write $Ba={\sum }_{j=1}^n{a}_j{N}_j$, and let $Rf=(f(x_1),\dots ,f(x_n){)}^T$. We use the standard interpolation convention of distinct ordered sites $x_1<\cdots <x_n$. Together with $N_i(x_i)>0$, the Schoenberg–Whitney theorem makes the B-spline collocation matrix $A$ invertible. Equivalently, its invertibility is implicit in the existence of the interpolation operator for every data vector.

If distinctness is not understood, the printed positivity condition alone is insufficient. For the order-two basis on spline knots $1,2,3,4$, take $x_1=x_2=5/2$. Both diagonal B-spline values are $1/2>0$, but the two rows of $A$ coincide. Arbitrary data cannot then be interpolated uniquely. The norm statement below concerns the intended well-defined interpolation operator.

Spline interpolation determines the coefficient vector by $Aa=Rf$, and henceNonnegativity and the subpartition of unity for B-splines giveFor completeness, extend the finite spline knot sequence beyond both ends. For the full sequence the order-one interval indicators sum to one. Summing the Cox-de Boor recurrence and shifting the index in its second term combines the two coefficients of each lower-order B-spline to one, so induction gives partition of unity at every order. Our finite collection is a subset of that nonnegative collection, and thus has sum at most one. Repeated spline knots are handled by the standard zero-term convention or a knot limit. This argument controls the entire interval, not only the basic knot interval.

Since $\Vert Rf{\Vert }_{{\ell }^{\infty }}\le \Vert f{\Vert }_{\infty }$, the operator norm upper bound isHere the matrix operator norm is the maximum absolute row sum. For the lower bound, choose a row of $A^{-1}$ with maximum absolute row sum and a data vector $y$ whose components are the signs of that row's entries. Then $\Vert y{\Vert }_{{\ell }^{\infty }}=1$ andA continuous piecewise-linear function taking these values at the distinct ordered sites, and constant outside their range, has supremum norm one. Apply the given uniform-norm stability of a B-spline basis to its coefficient vector:ThereforeThe construction of the bounded continuous data extension is what permits the matrix norm to give a lower bound for a function-space operator norm.

For the uniform spline knots, translation reduces every degree-two basis function to a quadratic cardinal B-spline. From the Cox-de Boor recurrence, an order-three function takes the valuesAt the prescribed site $x_i=i+2$, only $N_i$ and, when $i<n$, $N_{i+1}$ have nonzero values. Hence the B-spline collocation matrix isThe shift satisfies $S^n=0$. A finite geometric expansion gives the exact inverseIts $i$th absolute row sum is $2(n-i+1)$, so $\Vert A^{-1}{\Vert }_{{\ell }^{\infty }}=2n$. Use the preceding bounds and the supplied stability constant $d_3=3$:Thus the linear growth of shifted quadratic spline interpolation is $\Theta (n)$, which in particular is $O(n)$. The lower bound, rather than the $O(n)$ upper bound alone, proves that these spline interpolation operators are not uniformly bounded.