Oblique projections and abstract splines

[摘要] Given a closed subspace J of a Hilbert space H and a bounded linear operator A is an element of L(H) which is positive, consider the set of all A-self-adjoint projections onto J: P(A, J) = {Q is an element of L(H):Q(2) = Q, Q(H) = J, 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 xi is an element of H, consider the set of (T, J) spline interpolants to xi: sp(T, J, xi) = {eta is an element of xi + J : parallel toTetaparallel to = min(sigmais an element ofJ) parallel toT(xi + sigma)parallel to} A strong relationship exists between P(A, J) and sp(T, J, xi). In fact, P(A, J) is not empty if and only if sp(T, J, xi) is not empty for every xi is an element of H. In this case, for any xi is an element of H/J it holds sp(T, J, xi) = {(1 - Q)xi : Q is an element of P(A, J)} and for any xi is an element of H, the unique vector of sp(T, J, xi) with minimal norm is (1 - P-A,P-J)xi, where PA-J is a distinguished element of P(A, J). These results offer a generalization to arbitrary operators of several theorems by de Boor, Atteia, Sard and others, which hold for closed range operators. (C) 2002 Elsevier Science (USA).

[发布日期] 2002-08-01