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

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.