Oblique Projections and Abstract Splines

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-selfadjoint projections onto S P(A,S) = {Q ∈ L(H) : Q2 = Q , Q(H) = S , AQ = Q∗A}. In addition, if H1 is another Hilbert space, T : H → H1 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) = {η ∈ ξ + S : ‖Tη‖ = min σ∈S ‖T (ξ + σ)‖}. A strong relationship exists between P(A,S) and sp (T,S, ξ). In fact, P(A,S) is not enpty if and only if sp (T,S, ξ) is not empty for every ξ ∈ H. In this case, for any ξ ∈ H \ S it holds sp (T,S, ξ) = {(1−Q)ξ : Q ∈ P(A,S)} and for any ξ ∈ H the unique vector of sp (T,S, ξ) with minimal norm is (1−PA,S)ξ, where PA,S is a distiguished 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.