Oblique Projections and Abstract Splines
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...
| Autores: | , , |
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2002 |
| País: | Argentina |
| Recursos: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/110824 |
| Acesso em linha: | http://hdl.handle.net/11336/110824 |
| Access Level: | acceso abierto |
| Palavra-chave: | oblique projection spline https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Resumo: | 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: sP(T,S,ξ)= {n∈ξ +S:∥Tn∥=min_{σ∈s} ∥T(ξ + σ)∥}. A strong relationship exists between P(A, S) and s p(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. |
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