Related papers: On the best coapproximation problem in $\ell_1^n$

The paper gives some results on best proximity and fixed point for a class of generalized hybrid cyclic self-mappings in Banach spaces.

Several new characterizations of Banach spaces containing a subspace isomorphic to $\ell^1$, are obtained. These are applied to the question of when $\ell^1$ embeds in the injective tensor product of two Banach spaces.

In the problem of the longest common substring with $k$ mismatches we are given two strings $X, Y$ and must find the maximal length $\ell$ such that there is a length-$\ell$ substring of $X$ and a length-$\ell$ substring of $Y$ that differ…

We prove that the existence of best coapproximation to any element of the normed linear space out of any one dimensional subspace and its coincidence with the best approximation to that element out of that subspace characterizes a real…

Let $(e_i)$ be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are…

We show that $\Ext^2(\ell_2, \ell_2)\neq 0$ in the category of Banach spaces. This solves a sharpened version of Palamodov's problem and provides a solution to the second order version of Palais problem. We also show that $\Ext^2(\ell_1,…

Using a recently developed $\mathcal H$-calculus we propose a unified approach to the study of rational approximations of holomorphic semigroups on Banach spaces. We provide unified and simple proofs to a number of basic results on…

This paper addresses a multi-scale finite element method for second order linear elliptic equations with arbitrarily rough coefficient. We propose a local oversampling method to construct basis functions that have optimal local…

For Banach spaces $X,Y,$ we consider a distance problem in the space of bounded linear operators $\mathcal{L}(X,Y).$ Motivated by a recent paper \cite{RAO21}, we obtain sufficient conditions so that for a compact operator…

We consider $\ell_p$-direct sums ($1\leq p<\infty$) and $c_0$-direct sums of countably many normed spaces and find the duals of these spaces. We characterize the support functionals of arbitrary elements in these spaces to characterize…

The main purposes of this paper are (1) To survey the area of coarse embeddability of metric spaces into Banach spaces, and, in particular, coarse embeddability of different Banach spaces into each other; (2) To present new results on the…

While it is well known that nonlinear methods of approximation can often perform dramatically better than linear methods, there are still questions on how to measure the optimal performance possible for such methods. This paper studies…

This article explores anti-coproximinal and strongly anti-coproximinal subspaces in the spaces of vector-valued continuous functions and operator spaces. We provide a complete characterization of strongly anti-coproximinal subspaces in $…

We study finite subsets of $\ell_p$ and show that, up to nowhere dense and Haar null complement, all of them embed isometrically into any Banach space that uniformly contains the spaces $\ell_p^n$, $n \in \mathbb{N}$.

Our work is related to problems $73$ and $74$ of Mazur and Orlicz in ``The Scottish Book" (ed. R. D. Mauldin). Let $k_1, \ldots, k_n$ be nonnegative integers such that $\sum_{i=1}^{n} k_{i}=m$, and let $\mathbb{K}(k_1, \ldots, k_n; X)$,…

This note shows the existence of a sharp bilinear estimate for the Bourgain-type space and gives its application to the optimal local well/ill-posedness of the Cauchy problem for the Benjamin equation.

We show that there exists a strong uniform embedding from any proper metric space into any Banach space without cotype. Then we prove a result concerning the Lipschitz embedding of locally finite subsets of $\mathcal{L}_{p}$-spaces. We use…

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact space, Gr\"ochenig in \cite{G} discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all $1\le p\le\infty$, we…

We study the relationship between the point-wise symmetry of Birkhoff-James orthogonality and the geometry of the space of operators $\mathbb{B}(\ell_\infty^n,\ell_1^m)$. We show that any non-zero left-symmetric point in this space is a…

We study an infinite system of ordinary differential equations that models the evolution of coagulating and fragmenting clusters, which we assume to be composed of identical units. Under very mild assumptions on the coefficients we prove…