Related papers: Lower-Order Refinements of Greedy Approximation
We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity.…
Tsirelson's space $\mathcal{T}$ made its appearance in Banach space theory in 1974 soon to become one of the most significant counterexamples in the theory. Its structure broke the ideal pattern that analysts had conceived for a generic…
When developing robust preconditioners for multiphysics problems, fractional functions of the Laplace operator often arise and need to be inverted. Rational approximation in the uniform norm can be used to convert inverting those fractional…
We consider classes of objective functions of cardinality constrained maximization problems for which the greedy algorithm guarantees a constant approximation. We propose the new class of $\gamma$-$\alpha$-augmentable functions and prove…
Greedy algorithms which use only function evaluations are applied to convex optimization in a general Banach space $X$. Along with algorithms that use exact evaluations, algorithms with approximate evaluations are treated. A priori upper…
A lot of recent activity in the theory of cluster algebras has been directed towards various constructions of "natural" bases in them. One of the approaches to this problem was developed several years ago by P.Sherman - A.Zelevinsky who…
In this paper, our main focus is expressing real numbers on the non-integer bases. We denote those bases as $\beta$'s, which is also a real number and $\beta \in (1,2)$. This project has 3 main parts. The study of expansions of real numbers…
Let $0 < \theta \leqslant 1$. A sequence of positive integers $(b_n)_{n=1}^\infty$ is called a weak greedy approximation of $\theta$ if $\sum_{n=1}^{\infty}1/b_n = \theta$. We introduce the weak greedy approximation algorithm (WGAA), which,…
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ satisfying some mild assumptions and let $\alpha\in(0,n)$ and $\beta\in(1,\infty)$. In this article, when $\alpha\in(0,1)$, the authors first find a reasonable version…
In this paper, we introduce the notions of $\alpha$-quasicomplemented and totally $\alpha$-quasicomplemented subspaces and we established some results under these contexts. We show, for example, that if $X$ is a separable or reflexive…
Garling sequence spaces admit a renorming with respect to which their standard unit vector basis is 1-greedy. We also discuss some additional properties of these Banach spaces related to uniform convexity and superreflexivity. In…
We give a complete characterization of $2\pi$-periodic weights $w$ for which the usual trigonometric system forms a quasi-greedy basis for $L^p(\bT;w)$, i.e., bases for which simple thresholding approximants converge in norm. The…
The contextual bandit literature has traditionally focused on algorithms that address the exploration-exploitation tradeoff. In particular, greedy algorithms that exploit current estimates without any exploration may be sub-optimal in…
Let $\mathfrak{A}$ be a Banach algebra, and $\mathcal{X}$ a Banach $\mathfrak{A}$-bimodule. A bounded linear mapping $\mathcal{D}:\mathfrak{A}\rightarrow \mathcal{X}$ is approximately semi-inner derivation if there eixist nets…
We present various estimates for the Lebesgue constants of the thresholding greedy algorithm, in the case of general bases in Banach spaces. We show the optimality of these estimates in some situations. Our results recover and slightly…
In this paper, we establish new advances in the theory started by T. Oikhberg in [15] where the author joins greedy approximation theory with the use of sequences with gaps. Concretely, we address and partially answer three open questions…
We study $(\alpha,\beta)$-spanners for weighted graphs. We propose a simple greedy completion procedure which starts from a sparse initial graph, and repeatedly fixes pairs of vertices with a bad stretch, generalizing Kunedsen's additive…
We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements…
Expansion of real numbers is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we begin to systematically study expansions in multiple given bases in a reasonable way, which is a…
This paper is a follow up to the previous author's paper on convex optimization. In that paper we began the process of adjusting greedy-type algorithms from nonlinear approximation for finding sparse solutions of convex optimization…