Oblique Projections and Abstract Splines Original Research Article

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): 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: s p(T,S,ξ)=η∈ξ+S:∣∣Tη∣∣=minσ∈S∣∣T(ξ+σ)∣∣. A strong relationship exists between P(A,S) and sp(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−PA,S)ξ, where PA,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.