Vector Measure Orthonormal Systems and Self-weighted Functions Approximation

Abstract

If λ is a positive vector measure on l 2 , the notion of λ-orthonormal system of functions leads to a natural generalization of the relation between orthogonality and best approximation in Hilbert spaces for spaces L 2 (λ) of square integrable functions with respect to λ. We provide a vector orthogonality criterion that induces the definition of a particular projection on a subspace of L 2 (λ) that we call the selfweighted approximation. As an application, we show a new extrapolation technique. §1. Introduction and Basic ResultsThe properties of the spaces of integrable functions with respect to a vector measure have been studied in both abstract and applied contexts (see for example [1,4,11,3,10,13]). Following the investigation about applications of vector measure integration and the spaces of integrable functions with respect to vector measures (see [3,10,11,13]), we propose in this paper a general framework for function approximation. Our aim is to extend the notion of orthogonal projection of a function on a subspace of a Hilbert space to the context of the spaces L 2 (λ) of square integrable functions with respect to a vector measure λ in order to obtain new fitting procedures. Thus, Section 1 is devoted to introduce several definitions involving generalized orthonormal sequences and to obtain the main results on approximation with respect to

Orthogonal systems in $L^2$ spaces of a vector measure

“…The notion of m-orthonormal sequence is the natural generalization of the concept of orthonormal sequence in a Hilbert space L 2 (µ) and has been studied in [6,7,8,13,15]. We consider a sequence of real functions…”

“…This notion is defined by imposing simultaneously orthogonality with respect to all the elements of the family of scalar measures defined by the vector measure. This kind of orthogonality with respect to a vector measure has been studied in a series of papers (see [6,7,8,15]). In these papers we can find several examples (see Examples 4,5,and 10 [15]).…”

Vector measure orthogonal sequences in spaces of square integrable functions

“…In particular, the geometry of the strongly m-orthogonal sequences is well known and can be found in the papers [34,48,59]. Some applications on function approximation of m-orthogonal sequences were developed also in the papers [34,35,36].…”

“…The representation theorem for 2-convex order continuous Banach lattices with a weak unit establishes that such an space can be al- spaces L 2 (m) of 2-integrable functions has been defined an studied in the last ten years, see for instance ( [35,36]). In this memoir we consider the three notions of orthogonality with respect to a vector measure, the weak orthogonality, the orthogonality itself and the strongly version of it that we will formalize in the following sections.…”

“…Given a vector measure m : Σ → X, consider a sequence of (non zero) real functions {f n } n that are square m-integrable. We say that it is orthogonal with respect to m if for theory, orthogonality with respect to a vector measure has been studied in a series of papers in the last 10 years (see [34,35,36,59]). We define now formally the notion of m−orthogonality with respect to a vector measure.…”