Oblique Projections and Abstract Splines

Given a closed subspace of a Hilbert space and a bounded linear operator AL() which is positive, consider the set of all A-self-adjoint projections onto :

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In addition, if ₁ is another Hilbert space, T:→₁ is a bounded linear operator such that T^*T=A and ξ, consider the set of (T,) spline interpolants to ξ:

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A strong relationship exists between (A,) and sp(T,,ξ). In fact, (A,) is not empty if and only if s p(T,,ξ) is not empty for every ξ. In this case, for any ξ it holds

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and for any ξ, the unique vector of s p(T,,ξ) with minimal norm is (1−P_A,)ξ, where P_A, is a distinguished element of (A,). These results offer a generalization to arbitrary operators of several theorems by de Boor, Atteia, Sard and others, which hold for closed range operators.