Albiac Alesanco, Fernando José
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Albiac Alesanco
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Fernando José
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Estadística, Informática y Matemáticas
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InaMat2. Instituto de Investigación en Materiales Avanzados y Matemáticas
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Publication 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 MatematikaWe 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.Publication 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 MatemáticasOur 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).Publication 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 MatematikaIt 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.Publication 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 MatematikaFor 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