Oblique Projections and Abstract Splines

Abstract

Given a closed subspace S of a Hilbert space H and a bounded linear operator A ∈ L (H) which is positive, consider the set of all A-self-adjoint projections onto S: P(A,S) ={Q ∈ L(H) : Q^2 = Q, Q(H)=S, AQ = Q*A} In addition, if H_1 is another Hilbert space, T :H→H_1 is a bounded linear operator such that T*T= A and ξ ∈ H, consider the set of (T ,S) spline interpolants to ξ: sP(T,S,ξ)= {n∈ξ +S:∥Tn∥=min_{σ∈s} ∥T(ξ + σ)∥}. A strong relationship exists between P(A, S) and s p(T, S, ξ). In fact, P(A, S) is not empty if and only if s p(T, S, ξ) is not empty for every ξ ∈ H. In this case, for any ξ ∈ H\S it holds s p(T, S, ξ) = {(1 - Q)ξ:Q ∈ P(A, S)} and for any ξ ∈ H, the unique vector of s p(T, S, ξ) with minimal norm is (1 - P_A,S)ξ, where P_A,S is a distinguished element of P(A, S). These results offer a generalization to arbitrary operators of several theorems by de Boor, Atteia, Sard and others, which hold for closed range operators.