Abstract:

This doctoral thesis is devoted to the analysis of orthogonal sequences in subspaces of spaces L2(m) of square integrable functions with respect to a Banach space valued countably additive measure m. The motivation of our ...[+]
This doctoral thesis is devoted to the analysis of orthogonal sequences in subspaces of spaces L2(m) of square integrable functions with respect to a Banach space valued countably additive measure m. The motivation of our work is to generalize the geometric arguments that provide the classical approximation procedures in Hilbert spaces. The notion of orthogonality lies in the center of the Hilbert space theory, and it allows to develop the theory of convergence of sequences in these spaces. Almost everywhere convergence, norm convergence and weak convergence are nowadays well known topics in the Hilbert space function theory.
The Banach function spaces L2(m) of a vector measure m represent a broad class of Banach lattices: each 2convex order continuous Banach lattice with a weak unit can be represented (by means of an order isomorphism) as a space L2(m) for an adequate vector measure m. The integral structure that the
vector measure integration provides in these spaces allows to generalize the orthogonality arguments of the Hilbert space theory, although the spaces L2(m) are far from being Hilbert spaces.
The first chapter of this memoir is devoted to introduce some fundamental concepts on Banach function spaces, vector measure integration and other topics that will be necessary in the rest of the work. Some results on convergence of sequences in Banach function spaces and Banach spaces are explained, and the general framework is established. Some orthogonality arguments are already introduced, both for sequences in L2(m) and for the integrals of these sequences when the vector measure m is Hilbert space valued. Unconditional convergence for sequences from the abstract point of view of the function spaces of integrable functions is analyzed, and a version of the Kadec and Pelczynsky method for finding disjoint sequences for the vector measure setting is given. In the second chapter three notions of orthogonality of a sequence with respect to a vector measure are formally introduced, and the main characterizations
of these sequences are given. Weak morthogonal sequences, (natural)
morthogonal sequences and strongly morthogonal sequences are de ned and
studied, providing also examples that show the relation with some classical problems
in analysis. The geometry of these sets of sequences are also studied.
In Chapter 3 we analyze almost everywhere convergence of weakmorthogonal
sequences. Our main result is a general vector measure version of the Mencho 
Rademacher Theorem. A particular case involving c0sums of Hilbert spaces is
also intensively studied in order to show the properties of the convergence.
Finally, Chapter 4 is devoted to show a concrete application. We develop an
approximation method with respect to a parametric measure based on our ideas.
A Bochner integrable function and an weak morthonormal sequence are the
main elements of our procedure, that allows to nd the Fourier coefficients that
are in this case measurable functions for a given function in the space L2(m).
Some applications for signal approximation for data coming from experimental
acoustics are also shown.
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