Perfect Splines with Boundary Conditions of Least Norm Original Research Article

Let A = (aij)l,r−1i = 1, j = 0 and B = (bij) m,r−1i = 1,j = 0 be matrices of ranks l and m, respectively. Suppose that à = (( −1)jaij) ∈ SCl (sign consistent of order l) and B ∈ SCm. Denote by Pr,N(A, B; ν1, ..., νn) the set of perfect splines with N knots which have n distinct zeros in (0, 1) with multiplicities ν1, ..., νn, respectively. and satisfy AP(0) = 0, BP(1) = 0, where P(a) = (p(a), ..., P(r−1)(a))T. We show that there is a unique P*∈Pr,N(A, B; ν1, ..., νn) of least uniform norm and that P* is characterized by the equioscillatory property. This is closely related to the optimal recovery of smooth functions satisfying boundary conditions by using the Hermite data.