On a theorem of Nikol'skii
We present Bernstein lethargy theorem and examine the relationship between Bernstein lethargy theorem and reflexivity.
math.FA↗
arXiv subjects
Asuman G. Aksoy
Publications and source records attributed to Asuman G. Aksoy.
Representation Theorems of $\mathbb{R}$-trees and Brownian Motions Indexed by $\mathbb R$-trees
We provide a new representation of an $\mathbb R$-tree by using a special set of metric rays. We have captured the four-point condition from these metric rays and shown an equivalence between the $\mathbb R$-trees with radial and river metrics, and these sets of metric rays. In stochastic analysis, these graphical representation theorems are of particular interest in identifying Brownian motions indexed by $\mathbb R$-trees.
math.MG↗
Constructing an Element of a Banach Space with Given Deviation from its Nested Subspaces
This paper contains two improvements on a theorem of S. N. Bernstein for Banach spaces. We show that if $X$ is an arbitrary infinite-dimensional Banach space, $\{Y_n\}$ is a sequence of strictly nested subspaces of $ X$ and if $\{d_n\}$ is a non-increasing sequence of non-negative numbers tending to 0, then for any $c\in(0,1]$ we can find $x_{c} \in X$, such that the distance $ρ(x_{c}, Y_n)$ from $x_{c}$ to $Y_n$ satisfies $$ c d_n \leq ρ(x_{c},Y_n) \leq 4c d_n,~\mbox{for all $n\in\mathbb N$}. $$ We prove the above inequality by first improving Borodin (2006)'s result for Banach spaces by weakening his condition on the sequence $\{d_n\}$. The weakened condition on $d_n$ requires refinement of Borodin's construction to extract an element in $X$, whose distances from the nested subspaces are precisely the given values $d_n$.
math.FA↗
Minimal Projections with respect to Numerical Radius
In this paper we survey some results on minimality of projections with respect to numerical radius. We note that in the cases $L^p$, $p=1,2,\infty$, there is no difference between the minimality of projections measured either with respect to operator norm or with respect to numerical radius. However, we give an example of a projection from $l^p_3$ onto a two-dimensional subspace which is minimal with respect to norm, but not with respect to numerical radius for $p\neq 1,2,\infty$. Furthermore, utilizing a theorem of Rudin and motivated by Fourier projections, we give a criterion for minimal projections, measured in numerical radius. Additionally, some results concerning strong unicity of minimal projections with respect to numerical radius are given.
math.FA↗
The Apple Doesn't Fall Far From the (Metric) Tree: The Equivalence of Definitions
In this paper we prove the equivalence of definitions for metric trees and for δ-hyperbolic spaces. We point out how these equivalences can be used to understand the geometric and metric properties of δ-hyperbolic spaces and its relation to CAT(κ) spaces.
math.MG↗