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: s p(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 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.