Related papers: A remark on approximation with polynomials and gre…

We prove that if $\mathcal{X}$ is a quasi-greedy Markushevich basis of a Banach space $\mathbb{X}$, its dual basis $\mathcal{X}^*$ spans a norming subspace of $\mathbb{X}^*$. We also prove this result for weaker forms of quasi-greediness,…

In [25], T. Oikhberg introduced and studied variants of the greedy and weak greedy algorithms for sequences with gaps, with a focus on the $\mathbf n$-$t$-quasi-greedy property that is based on them. Building upon this foundation, our…

The aim of this paper is to show that almost greedy bases induce tighter embeddings in superreflexive Banach spaces than in general Banach spaces. More specifically, we show that an almost greedy basis in a superreflexive Banach space…

This paper describes a simple greedy D-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards)…

Let $\mathcal{F}$ be a hereditary collection of finite subsets of $\mathbb{N}$. In this paper, we introduce and characterize $\mathcal{F}$-(almost) greedy bases. Given such a family $\mathcal{F}$, a basis $(e_n)_n$ for a Banach space $X$ is…

For a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space the associated conditionality constants $k_{m}[\mathcal{B}]$ verify the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$. Answering a question raised by Temlyakov, Yang,…

In 1999, S. V. Konyagin and V. N. Temlyakov introduced the so-called Thresholding Greedy Algorithm. Since then, there have been many interesting and useful characterizations of greedy-type bases in Banach spaces. In this article, we study…

We continue the study initiated in [F. Albiac and P. Wojtaszczyk, Characterization of $1$-greedy bases, J. Approx. Theory 138 (2006), no. 1, 65-86] of properties related to greedy bases in the case when the constants involved are sharp,…

We continue the study of Lebesgue-type parameters for various greedy algorithms in quasi-Banach spaces. First, we introduce a parameter that can be used with the quasi-greedy parameter to obtain the exact growth of the Lebesgue parameter…

The fact that finite direct sums of two or more mutually different spaces from the family $\{\ell_{p} : 1\le p<\infty\}\cup c_{0}$ fail to have greedy bases is stated in [Dilworth et al., Greedy bases for Besov spaces, Constr. Approx. 34…

We prove that several forms of the Bernstein polynomials with integer coefficients possess the property of simultaneous approximation, that is, they approximate not only the function but also its derivatives. We establish direct estimates…

We introduce and study the notion of weak semi-greedy systems -which is inspired in the concepts of semi-greedy and Branch semi-greedy systems and weak thresholding sets-, and prove that in the context Markushevich bases in infinite…

We prove some results on the rate of convergence of greedy algorithms, which provide expansions. We consider both the case of Hilbert spaces and the more general case of Banach spaces. The new ingredient of the paper is that we bound the…

Reduced bases have been introduced for the approximation of parametrized PDEs in applications where many online queries are required. Their numerical efficiency for such problems has been theoretically confirmed in \cite{BCDDPW,DPW}, where…

The rate of convergence of the classical Thresholding Greedy Algorithm with respect to bases is studied in this paper. We bound the error of approximation by the product of both norms -- the norm of $f$ and the $A_1$-norm of $f$. We obtain…

In [Y.~K.~Hu, K.~A.~Kopotun, X.~M.~Yu, Constr. Approx. 2000], the authors have obtained a characterization of best $n$-term piecewise polynomial approximation spaces as real interpolation spaces between $L^p$ and some spaces of bounded…

We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…

We consider the $X$-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in $L_p[0,1]$…

We analyse the problem of controllability for parameter-dependent linear finite-dimensional systems. The goal is to identify the most distinguished realisations of those parameters so to better describe or approximate the whole range of…

The approximation of a discrete probability distribution $\mathbf{t}$ by an $M$-type distribution $\mathbf{p}$ is considered. The approximation error is measured by the informational divergence $\mathbb{D}(\mathbf{t}\Vert\mathbf{p})$, which…