Oblique projections and abstract splines

Abstract

Given a closed subspace L 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 Y: ℘(A,Y) = {Q ∈ L(ℋ): Q2 = Q, Q(ℋ) = Y, 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, Y) spline interpolants to ξ: sp(T, Y, ξ) = { η ε ξ + Y : ∥Tη∥ = min ∥T(ξ + σ)∥}. A strong relationship exists between ℘(A, Y) and s p(T, Y, ξ). In fact, ∥(A, Y) is not empty if and only if s p(T, Y, ξ) is not empty for every ξ ∈ ℋ. In this case, for any ξ ∈ ℋ\Y it holds s p(T, Y, ξ) = {(1 - Q)ξ:Q ∈ ℘(A, Y)} and for any ξ ∈ ℋ, the unique vector of s p(T, Y, ξ) with minimal norm is (1 - PA,Y)ξ, where PA,L is a distinguished element of ℘(A, Y). These results offer a generalization to arbitrary operators of several theorems by de Boor, Atteia, Sard and others, which hold for closed range operators.