On uncountable unconditional bases in Banach spaces
On τδ-completeness of H(U).
On ω*-basic sequences and their applications to the study of Banach spaces
Ondelettes et bases hilbertiennes.
Ondelettes et fonctions splines
Opérateurs diagonaux dans les espaces à bases.
Operator ideal properties of vector measures with finite variation
Given a vector measure m with values in a Banach space X, a desirable property (when available) of the associated Banach function space L¹(m) of all m-integrable functions is that L¹(m) = L¹(|m|), where |m| is the [0,∞]-valued variation measure of m. Closely connected to m is its X-valued integration map Iₘ: f ↦ ∫f dm for f ∈ L¹(m). Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property L¹(m) = L¹(|m|) and the...
Ordinal indices and Ramsey dichotomies measuring c₀-content and semibounded completeness
We study the c₀-content of a seminormalized basic sequence (χₙ) in a Banach space, by the use of ordinal indices (taking values up to ω₁) that determine dichotomies at every ordinal stage, based on the Ramsey-type principle for every countable ordinal, obtained earlier by the author. We introduce two such indices, the c₀-index and the semibounded completeness index , and we examine their relationship. The countable ordinal values that these indices can take are always of the form . These results...
Ordinal indices in Banach spaces.
Orlicz sequence spaces with a unique spreading model.
Orthogonal bases in 1∞ (X).
Orthonormal bases for -adic continuous and continuously differentiable functions
Orthonormal bases for spaces of continuous and continuously differentiable functions defined on a subset of Zp.
Let K be a non-Archimedean valued field which contains Qp, and suppose that K is complete for the valuation |·|, which extends the p-adic valuation. Vq is the closure of the set {aqn | n = 0,1,2,...} where a and q are two units of Zp, q not a root of unity. C(Vq --> K) (resp. C1(Vq --> K)) is the Banach space of continuous functions (resp. continuously differentiable functions) from Vq to K. Our aim is to find orthonormal bases for C(Vq --> K) and C1(Vq --> K).
-local unconditional structure of Banach spaces
Partial unconditionality of weakly null sequences.
We survey a combinatorial framework for studying subsequences of a given sequence in a Banach space, with particular emphasis on weakly-null sequences. We base our presentation on the crucial notion of barrier introduced long time ago by Nash-Williams. In fact, one of the purposes of this survey is to isolate the importance of studying mappings defined on barriers as a crucial step towards solving a given problem that involves sequences in Banach spaces. We focus our study on various forms of ?partial...
Partially bounded sets of infinite width.
Perturbations of Schauder bases in the spaces C(K) and , p<1
p-Frames and Shift Invariant Subspaces of L...
Pointwise smoothness, two-microlocalization and wavelet coefficients.
In this paper we shall compare three notions of pointwise smoothness: the usual definition, J.M. Bony's two-microlocal spaces Cx0s,s', and the corresponding definition on the wavelet coefficients. The purpose is mainly to show that these two-microlocal spaces provide "good substitutes" for the pointwise Hölder regularity condition; they can be very precisely compared with this condition, they have more functional properties, and can be characterized by conditions on the wavelet coefficients. We...
Positive bases in ordered subspaces with the Riesz decomposition property
In this article we suppose that E is an ordered Banach space whose positive cone is defined by a countable family of positive continuous linear functionals on E, i.e. E₊ = x ∈ E | for each i, and we study the existence of positive (Schauder) bases in ordered subspaces X of E with the Riesz decomposition property. We consider the elements x of E as sequences and we develop a process of successive decompositions of a quasi-interior point of X₊ which at each step gives elements with smaller support....