Optimal inverses and abstract splines

Global solutions of approximation problems in Hilbert spaces

“…The following result relates the existence of a W -optimal inverse to the existence of a solution of (OSMP). Some equivalences of the next proposition were proven in [10,Theorem 4.2] for V ∈ L(H, F ) with closed range. The proofs of such equivalences are included in order to remark that the range of V need not be closed.…”

“…To study the existence of solutions of inconsistent linear systems under seminorms defined by positive semidefinite matrices, Mitra defined the optimal inverses for matrices [23]. In [10], Mitra's concept was extended to Hilbert spaces:…”

“…It can be seen that R(V * ) = R(V * V ) ⊆ R(T * T + V * V ) if and only if R(T * T + V * V ) = R(T * T ) + R(V * V ). But this is equivalent to (T * T, N (V )) being compatible, see [10,Theorem 3.2].…”

Total least squares problems on infinite dimensional spaces

“…We present necessary conditions for a pair (A 0 , x 0 ), to be a solution of the RWTLS problem. We observe that if the RWTLS problem has a solution (A 0 , x 0 ), then x 0 is a solution of the classical smoothing problem [3,9,10,13,14] and the results obtained in [11] can be applied for giving necessary conditions for the existence of solution of problem (1.2). In section 3.3, we study the case where the regularization is given by a multiple of the identity operator, ρI.…”

The minus order and range additivity

“…Given two linear bounded operators A and B acting on a Hilbert space , we say that A and B have the range additivity property if R(A + B ) = R(A) + R(B ), where R(T ) stands for the range of an operator T . Operators with this property have been studied in [4] and [5] (see also [10]). Recall that if A and B are two bounded linear Hilbert space operators then A − ≤ B (where the symbol " − ≤" stands for the minus order of operators) if and only if there are oblique projections P and Q such that A = P B and A * = Q B * .…”