Consider the interpolation problem of Theorem 20.2. Show that                             Bj,M,u(tj ) = 0 for some j ∈ {−M,...,K − 1}  implies that the solution of the interpolation problem is not...

Consider the interpolation problem of Theorem 20.2. Show that

                            Bj,M,u(tj ) = 0 for some j ∈ {−M,...,K − 1}

 implies that the solution of the interpolation problem is not unique.

Hint: First prove the assertion for M = 0. Then assume M > 0 and consider the cases tj ≤ uj and tj ≥ uj+M+1. Show that tj ≤ uj implies Bk,M,u(ti)=0 for i ≤ j and k ≥ j. Conclude that in this case the matrix (Bk,M,u(ti))i,k is not regular. Argue similarly in the case tj ≥ uj+M+1.