Open Access
Unconditional and quasi-greedy bases in L-p with applications to Jacobi polynomials Fourier series
(European Mathematical Society, 2019) Albiac Alesanco, Fernando José; Ansorena, José L.; Ciaurri, Óscar; Varona, Juan L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
We show that the decreasing rearrangement of the Fourier series with respect to the Jacobi polynomials for functions in L-p does not converge unless p = 2. As a by-product of our work on quasi-greedy bases in L-p(µ), we show that no normalized unconditional basis in L-p, p not equal 2, can be semi-normalized in L-q for q not equal p, thus extending a classical theorem of Kadets and Pelczynski from 1968.
Open Access
Asymptotic greediness of the Haar system in the spaces Lp[0 , 1] , 1< p< ∞
(Springer, 2019) Albiac Alesanco, Fernando José; Ansorena, José L.; Berná, Pablo M.; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant Cg[H(p), Lp] of the (normalized) Haar system H(p) in Lp[0 , 1] for 1 < p < ∞. We will show that the super-democracy constant of H(p) in Lp[0 , 1] grows as p∗= max { p, p/ (p- 1) } as p∗ goes to ∞. Thus, since the unconditionality constant of H(p) in Lp[0 , 1] is p∗- 1 , the well-known general estimates for the greedy constant of a greedy basis obtained from the intrinsic features of greediness (namely, democracy and unconditionality) yield that p∗≲Cg[H(p),Lp]≲(p∗)2. Going further, we develop techniques that allow us to close the gap between those two bounds, establishing that, in fact, Cg[H(p), Lp] ≈ p∗. Our work answers a question that was raised by Hytonen (2015).
Open Access
Building highly conditional almost greedy and quasi-greedy bases in Banach spaces
(Elsevier, 2019) Albiac Alesanco, Fernando José; Ansorena, José L.; Dilworth, S. J.; Kutzarova, Denka; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
It is known that for a conditional quasi-greedy basis B in a Banach space X, the associated sequence (k(m)[B](m=1)(infinity) of its conditionality constants verifies the estimate k(m)[B] = O(log m) and that if the reverse inequality log m =O(k(m)[B]) holds then X is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate k(m)[B] =O(log m)(1-epsilon) for some epsilon > 0. However, in the existing literature one finds very few instances of spaces possessing quasi-greedy basis with conditionality constants "as large as possible." Our goal in this article is to fill this gap. To that end we enhance and exploit a technique developed by Dilworth et al. in [15] and craft a wealth of new examples of both non-superreflexive classical Banach spaces having quasi-greedy bases B with k(m)[B] = O(log m) and superreflexiye classical Banach spaces having for every epsilon > 0 quasi-greedy bases B with k(m)[B] = O(log m)(1-epsilon). Moreover, in most cases those bases will be almost greedy.
Open Access
Conditional quasi-greedy bases in non-superreflexive Banach spaces
(Springer, 2019) Albiac Alesanco, Fernando José; Ansorena, José L.; Wojtaszczyk, Przemyslaw; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
For a conditional quasi-greedy basis B in a Banach space, the associated conditionality constants km[B] verify the estimate km[B]=O(logm). Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies km[B]=O((logm)1-E) for some 0
Open Access
The Tsirelson space tau (p) has a unique unconditional basis up to permutation for 0 < p < 1
(Hindawi Publishing Corporation, 2009) Albiac Alesanco, Fernando José; Leránoz Istúriz, María Camino; Matemáticas; Matematika
We show that the p-convexified Tsirelson space tau((p)) for 0 < p < 1 and all its complemented subspaces with unconditional basis have unique unconditional basis up to permutation. The techniques involved in the proof are different from the methods that have been used in all the other uniqueness results in the nonlocally convex setting. Copyright (C) 2009 F. Albiac and C. Leranoz.
Open Access
On perfectly homogeneous bases in quasi-Banach spaces
(Hindawi Publishing Corporation, 2009) Albiac Alesanco, Fernando José; Leránoz Istúriz, María Camino; Matemáticas; Matematika
For 0 < p < infinity the unit vector basis of l(p) has the property of perfect homogeneity: it is equivalent to all its normalized block basic sequences, that is, perfectly homogeneous bases are a special case of symmetric bases. For Banach spaces, a classical result of Zippin (1966) proved that perfectly homogeneous bases are equivalent to either the canonical c(0)-basis or the canonical l(p)-basis for some 1 <= p < infinity. In this note, we show that (a relaxed form of) perfect homogeneity characterizes the unit vector bases of l(p) for 0 < p < 1 as well amongst bases in nonlocally convex quasi-Banach spaces. Copyright (C) 2009 F. Albiac and C. Leranoz.