Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 61 / 4

Use the standard partition normalization of B-splines, with order $k$ meaning degree $k-1$. The Cox-de Boor recurrence on arbitrary increasing spline knots isstarting with $N_{i,1}(x)=1_{[t_i,t_{i+1})}(x)$. Terms with zero denominator are interpreted as zero in the usual repeated-knot convention. For the uniform knots here, $t_i=i$, this reduces toThe index of the second lower-order B-spline is $i+1$, not $i-1$. Both lower-order supports combine to give $[i,i+k]$.

Uniform knots give translation invariance $N_{i,k}(x)=N_{0,k}(x-i)$. Put $v_{k,j}=N_{0,k}(j)$ and set missing values to zero. The recurrence in part (a) gives the exact numerical recursionThe order-two starting row is $(0,1,0)$. For example, the order-four middle value is $(2\cdot \frac{1}{2}+2\cdot \frac{1}{2})/3=2/3$, and the order-six central value is $(3\cdot \frac{11}{24}+3\cdot \frac{11}{24})/5=66/120$. Iteration gives the requested array, listing $N_{0,k}(0),\dots ,N_{0,k}(k)$:The symmetry follows from reflection of the uniform Cardinal B-spline about the midpoint of its support. The row sums equal one, consistent with sampling the uniform partition of unity. These are partition-normalized values; normalizing each spline to have peak one would produce a different array.

For order six, the support-midpoint sites are $x_i=i+3$, soBy part (b), the complete finite matrix is the symmetric five-diagonal matrixOnly indices from $0$ through $n$ are retained; the first and last rows are truncated, with no exterior coefficients or wraparound terms. For a row with all possible neighbors, the strict diagonal dominance margin is $(66-26-26-1-1)/120=1/10$. Boundary rows have an equal or larger margin.

Here is the appropriate inverse estimate with its proof. If a square matrix $A$ has $|a_{ii}|-{\sum }_{j\ne i}|a_{ij}|\ge d>0$, choose an index $i$ maximizing $|z_i|$. The reverse triangle inequality givesThus $Az=0$ implies $z=0$, and a square finite matrix is invertible. Applying the same estimate with $z=A^{-1}b$ proves $\Vert A^{-1}b{\Vert }_{{\ell }^{\infty }}\le d^{-1}\Vert b{\Vert }_{{\ell }^{\infty }}$. The inverse infinity-norm bound from diagonal dominance consequently yieldsThis also proves existence and uniqueness of the requested interpolating spline for every data vector and every finite $n$. The quintic cardinal spline midpoint collocation matrix includes the cases of one or two coefficients, where the dominance margin is even larger.