Quasi-greedy Bases Research Articles

Linear versus nonlinear forms of partial unconditionality of bases

The main results in this paper contribute to bringing to the fore novel underlying connections between the contemporary concepts and methods springing from greedy approximation theory with the well-established techniques of classical Banach spaces. We do that by showing that bounded-oscillation unconditional bases, introduced by Dilworth et al. in 2009 in the setting of their search for extraction principles of subsequences verifying partial forms of unconditionality, are the same as truncation quasi-greedy bases, a new breed of bases that appear naturally in the study of the performance of the thresholding greedy algorithm in Banach spaces. We use this identification to provide examples of bases that exhibit that bounded-oscillation unconditionality is a stronger condition than Elton's near unconditionality. We also take advantage of our arguments to provide examples that allow us to tell apart certain types of bases that verify either debilitated unconditionality conditions or weaker forms of quasi-greediness in the context of abstract approximation theory.

Democracy of quasi-greedy bases in -Banach spaces with applications to the efficiency of the Thresholding Greedy Algorithm in the Hardy spaces

We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0< p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p({\mathbb {D}})$ for $0< p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p({\mathbb {D}}^d)$ for $d\ge 2$ is, solving thus a problem that was raised in [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.

Bidemocratic Bases and Their Connections with Other Greedy-Type Bases

In nonlinear greedy approximation theory, bidemocratic bases have traditionally played the role of dualizing democratic, greedy, quasi-greedy, or almost greedy bases. In this article we shift the viewpoint and study them for their own sake, just as we would with any other kind of greedy-type bases. In particular we show that bidemocratic bases need not be quasi-greedy, despite the fact that they retain a strong unconditionality flavor which brings them very close to being quasi-greedy. Our constructive approach gives that for each 1<p<infty the space ell _p has a bidemocratic basis which is not quasi-greedy. We also present a novel method for constructing conditional quasi-greedy bases which are bidemocratic, and provide a characterization of bidemocratic bases in terms of the new concepts of truncation quasi-greediness and partially democratic bases.

Weak forms of unconditionality of bases in greedy approximation

We study a new class of bases, weaker than quasi-greedy bases, which retain their unconditionality properties and can provide the same optimality for the thresholding greedy algorithm. We measure how far these bases are from being unconditional and use th

Quasi-greedy bases for sequences with gaps

In this paper, we establish new results in the theory started by T. Oikhberg in Oikhberg (2018) where the author joins greedy approximation theory with the use of sequences with gaps. Concretely, we study and provide some answers to three open questions related to quasi-greedy bases for sequences with gaps posed in Oikhberg (2018)[Section 6].

Quasi-greedy bases in ℓp (0 < p < 1) are democratic

The list of known Banach spaces whose linear geometry determines the (nonlinear) democracy functions of their quasi-greedy bases to the extent that they end up being democratic, reduces to c0, ℓ2, and all separable L1-spaces. Oddly enough, these are the only Banach spaces that, when they have an unconditional basis, it is unique. Our aim in this paper is to study the connection between quasi-greediness and democracy of bases in non-locally convex spaces. We prove that all quasi-greedy bases in ℓp for 0<p<1 (which also has a unique unconditional basis) are democratic with fundamental function of the same order as (m1/p)m=1∞. The methods we develop allow us to obtain even more, namely that the same occurs in any separable Lp-space, 0<p<1, with the bounded approximation property.

On certain subspaces of $${\ell _p}$$ for $${0&lt;p\le 1}$$ and their applications to conditional quasi-greedy bases in p-Banach spaces

We construct for each $$0<p\le 1$$ an infinite collection of subspaces of $$\ell _p$$ that extend the example of Lindenstrauss (Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 12, 539–542, 1964) of a subspace of $$\ell _{1}$$ with no unconditional basis. The structure of this new class of p-Banach spaces is analyzed and some applications to the general theory of $${\mathcal {L}}_{p}$$ -spaces for $$0<p<1$$ are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in p-Banach spaces for $$p<1$$ . Among the topics we consider are the existence of infinitely many conditional quasi-greedy bases in the spaces $$\ell _{p}$$ for $$p\le 1$$ and the careful examination of the conditionality constants of the “natural basis” of these spaces.

Unconditional and quasi-greedy bases in $L_p$ with applications to Jacobi polynomials Fourier series

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(\mu) , we show that no normalized unconditional basis in L_p , p \neq 2 , can be semi-normalized in L_q for q \neq p , thus extending a classical theorem of Kadets and Pełczyński from 1962.

Building highly conditional almost greedy and quasi-greedy bases in Banach spaces

It is known that for a conditional quasi-greedy basis B in a Banach space X, the associated sequence (km[B])m=1∞ of its conditionality constants verifies the estimate km[B]=O(log⁡m) and that if the reverse inequality log⁡m=O(km[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 km[B]=O(log⁡m)1−ϵ for some ϵ>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 km[B]=O(log⁡m) and superreflexive classical Banach spaces having for every ϵ>0 quasi-greedy bases B with km[B]=O(log⁡m)1−ϵ. Moreover, in most cases those bases will be almost greedy.

Lebesgue constants for Chebyshev thresholding greedy algorithms

We investigate the efficiency of Chebyshev Thresholding Greedy Algorithm (CTGA) for an n-term approximation with respect to general bases in a Banach space. We show that the convergence property of CTGA is better than TGA for non-quasi-greedy bases. Then we determine the exact rate of the Lebesgue constants L_{n}^{mathrm{ch}} for two examples of such bases: the trigonometric system and the summing basis. We also establish the upper estimates for L_{n}^{mathrm{ch}} with respect to general bases in terms of quasi-greedy parameter, democracy parameter and A-property parameter. These estimates do not involve an unconditionality parameter, therefore they are better than those of TGA. In particular, for conditional quasi-greedy bases, a faster convergence rate is obtained.

Embeddings and Lebesgue-Type Inequalities for the Greedy Algorithm in Banach Spaces

We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of not necessarily quasi-greedy bases.

Efficiency of weak greedy algorithms for m-term approximations

We investigate the efficiency of weak greedy algorithms for $m$-term expansional approximation with respect to quasi-greedy bases in general Banach spaces. We estimate the corresponding Lebesgue constants for the weak thresholding greedy algorithm (WTGA) and weak Chebyshev thresholding greedy algorithm. Then we discuss the greedy approximation on some function classes. For some sparse classes induced by uniformly bounded quasi-greedy bases of $L_p$, $1＜ p＜\infty$, we show that the WTGA realizes the order of the best $m$-term approximation. Finally, we compare the efficiency of the weak Chebyshev greedy algorithm (WCGA) with the thresholding greedy algorithm (TGA) when applying them to quasi-greedy bases in $L_p,\,1\leq p＜\infty$, by establishing the corresponding Lebesgue-type inequalities. It seems that when $p＞2$ the WCGA is better than the TGA.

On quasi-greedy bases associated with unitary representations of countable groups

On quasi-greedy bases associated with unitary representations of countable groups

Lorentz Spaces and Embeddings Induced by Almost Greedy Bases in Banach Spaces

The aim of this paper is to undertake a systematic qualitative study of the built-in symmetry of almost greedy bases in Banach spaces. More specifically, by refining the techniques that Wojtaszczyk used in J Approx Theory 107(2), 293–314 2000 for quasi-greedy bases in Hilbert spaces, we show that an almost greedy basis in a Banach space X naturally induces embeddings that allow sandwiching X between two symmetric sequence spaces. Using classical interpolation techniques in combination with duality, we also explore what we label as interpolation of greedy bases. It is then proved that the only almost greedy basis shared by any two \(\ell _{p}\) spaces is equivalent to the standard unit vector basis and that there is no basis which is simultaneously (normalized and) greedy in two different \(L_{p}\) spaces. As a by-product of our work, we obtain a new characterization of greedy bases in Banach spaces in terms of bounded linear operators.

Conditional quasi-greedy bases in Hilbert and Banach spaces

For quasi-greedy bases B in Hilbert spaces, we give an improved bound of the associated conditionality constants kN (B) = O(logN)1−e, for some e > 0, answering a question by Temlyakov. We show the optimality of this bound with an explicit construction, based on a refinement of the method of Olevskii. This construction leads to other examples of quasi-greedy bases with large kN in Banach spaces, which are of independent interest.

Lebesgue-Type Inequalities for Quasi-greedy Bases

We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N-term approximation ∥x−GNx∥ and the best N-term approximation σN(x) is controlled by max{μ(N),kN}, where μ(N) and kN are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is O(logN). We show with various examples that these bounds are actually attained.

Quasi-greedy bases and Lebesgue-type inequalities

Our primary interest in this chapter is in approximation in L p with respect to quasi-greedy bases. The presentation of this chapter is based on the recent paper [21].

Trigonometric bases for matrix weighted [formula omitted]-spaces

We give a complete characterization of 2 π-periodic matrix weights W for which the vector-valued trigonometric system forms a Schauder basis for the matrix weighted space L p ( T ; W ) . Then trigonometric quasi-greedy bases for L p ( T ; W ) are considered. Quasi-greedy bases are systems for which the simple thresholding approximation algorithm converges in norm. It is proved that such a trigonometric basis can be quasi-greedy only for p = 2 , and whenever the system forms a quasi-greedy basis, the basis must actually be a Riesz basis.

Greedy approximation with regard to non-greedy bases

The main goal of this paper is to understand which properties of a basis are important for certain direct and inverse theorems in nonlinear approximation. We study greedy approximation with regard to bases with different properties. We consider bases that are tensor products of univariate greedy bases. Some results known for unconditional bases are extended to the case of quasi-greedy bases.

Trigonometric Quasi-Greedy Bases for $L^p(T;w)$

Let B = {en}n∈N be a bounded Schauder basis for a Banach space X, i.e., a basis for which 0 ρ(j). Then the greedy m-term approximant to x is given by Gm(x) = ∑m j=1 e ∗ ρ(j)(x)eρ(j). The question is whether the greedy algorithm is convergent. This is clearly the case for an unconditional basis where the expansion x = ∑∞ k=1 e ∗ k(x)ek converges regardless of the ordering. However, Temlyakov and Konyagin [7] showed that the greedy algorithm may also converge for certain conditional bases. This lead them to define so-called quasi-greedy bases, see [7]. The definition of a quasi-greedy basis in [7] is slightly technical, but it was shown by Wojtaszczyk [11] to be equivalent to to the following statement which we use as definition.