Perfect Splines with Boundary Conditions of Least Norm

Abstract

Let A = ( a ij ) l, r−1 i = 1, j = 0 and B = ( b ij) m, r−1 i = 1, j = 0 be matrices of ranks l and m, respectively. Suppose that à = (( −1) j a ij ) ∈ SC l (sign consistent of order l) and B ∈ SC m . Denote by P r,N ( A, B; ν 1, ..., ν n) the set of perfect splines with N knots which have n distinct zeros in (0, 1) with multiplicities ν 1, ..., ν n, respectively. and satisfy A P (0) = 0, B P (1) = 0, where P ( a) = ( p( a), ..., P (r−1)( a)) T . We show that there is a unique P*∈ P r,N ( A, B; ν 1, ..., ν n) 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.