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Dogan Comez

Publications and source records attributed to Dogan Comez.

8 recordsLinked to original sources
arXiv2022

Canonical sequences of optimal quantization for condensation measures

We consider condensation measures of the form $P:=\frac 13 P\circ S_1^{-1}+ \frac 13 P\circ S_2^{-1}+ \frac 13 ν$ associated with the system $(\mathcal{S}, (\frac 13, \frac 13, \frac 13), ν) , $ where $\mathcal{S}=\{S_i\}_{i=1}^2 $ are contractions and $ ν$ is a Borel probability measure on $\mathbb R$ with compact support. Let $D(μ)$ denote the quantization dimension of a measure $μ$ if it exists. In this paper, we study self-similar measures $ν$ satisfying $D(ν)>κ$, $D(ν)<κ$, and $D(ν)=κ, $ respectively, where $κ$ is the unique number satisfying $[\frac13 (\frac{1}{5})^2]^{\fracκ{2+κ}}=\frac 12. $ For each case we construct two sequences $a(n)$ and $F(n)$, which are utilized in determining the optimal sets of $F(n)$-means and the $F(n)$th quantization errors for $P. $ We also show that for each measure $ν$ the quantization dimension $D(P)$ of $P$ exists and satisfies $D(P)=\max\{κ, D(ν)\}. $ Moreover, we show that for $D(ν)>κ$, the $D(P)$-dimensional lower and upper quantization coefficients are finite, positive and unequal; and for $D(ν)\leq κ$, the $D(P)$-dimensional lower quantization coefficient is infinity.

math.DS↗
arXiv2022

Quantization for infinite affine transformations

Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite set. In this article, we consider a probability distribution generated by an infinite system of affine transformations $\{S_{ij}\}$ on $\mathbb R^2$ with associated probabilities $\{p_{ij}\}$ such that $p_{ij}>0$ for all $i, j\in \mathbb N$ and $\sum_{i, j=1}^\infty p_{ij}=1$. For such a probability measure $P$, the optimal sets of $n$-means and the $n$th quantization error are calculated for every natural number $n$. It is shown that the distribution of such a probability measure is the same as that of the direct product of the Cantor distribution. In addition, it is proved that the quantization dimension $D(P)$ exists and is finite; whereas, the $D(P)$-dimensional quantization coefficient does not exist, and the $D(P)$-dimensional lower and the upper quantization coefficients lie in the closed interval $[\frac{1}{12}, \frac{5}{4}]$.

math.DS↗
arXiv2019

Quantization for uniform distributions of Cantor dusts on $\mathbb{R}^2$

Let $P$ be a Borel probability measure on $\mathbb R^2$ supported by the Cantor dusts generated by a set of $4^u,\ u\geq 1$, contractive similarity mappings satisfying the strong separation condition. For this probability measure, we determine the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$. In addition, it is shown that though the quantization dimension of the measure $P$ is known, the quantization coefficient for $P$ does not exist.

math.DS↗
arXiv2019

Individual ergodic theorems for infinite measure

Given a $σ$-finite infinite measure space $(Ω,μ)$, it is shown that any Dunford-Schwartz operator $T:\,\mathcal L^1(Ω)\to\mathcal L^1(Ω)$ can be uniquely extended to the space $\mathcal L^1(Ω)+\mathcal L^\infty(Ω)$. This allows to find the largest subspace $\mathcal R_μ$ of $\mathcal L^1(Ω)+\mathcal L^\infty(Ω)$ such that the ergodic averages $\frac1n\sum\limits_{k=0}^{n-1}T^k(f)$ converge almost uniformly (in Egorov's sense) for every $f\in\mathcal R_μ$ and every Dunford-Schwartz operator $T$. Utilizing this result, almost uniform convergence of the averages $\frac1n\sum\limits_{k=0}^{n-1}β_kT^k(f)$ for every $f\in\mathcal R_μ$, any Dunford-Schwartz operator $T$ and any bounded Besicovitch sequence $\{β_k\}$ is established. Further, given a measure preserving transformation $τ:Ω\toΩ$, Assani's extension of Bourgain's Return Times theorem to $σ$-finite measure is employed to show that for each $f\in\mathcal R_μ$ there exists a set $Ω_f\subsetΩ$ such that $μ(Ω\setminusΩ_f)=0$ and the averages $\frac1n\sum\limits_{k=0}^{n-1}β_kf(τ^kω)$ converge for all $ω\inΩ_f$ and any bounded Besicovitch sequence $\{β_k\}$. Applications to fully symmetric subspaces $E\subset\mathcal R_μ$ are given.

math.FA↗
arXiv2019

Quantization for uniform distributions on stretched Sierpiński triangles

In this paper, we have considered a uniform probability distribution supported by a stretched Sierpiński triangle. For this probability measure, the optimal sets of $n$-means and the $n$th quantization errors are determined for all $n\geq 2$. In addition, it is shown that the quantization coefficient for such a measure does not exist though the quantization dimension exists.

math.DS↗
arXiv2016

On pointwise ergodic theorems for infinite measure

For a Dunford-Schwartz operator in the $L^p-$space, $1\leq p< \infty$ , of an arbitrary measure space, we prove pointwise convergence of the conventional and Besicovitch weighted ergodic averages. Pointwise convergence of various types of ergodic averages in fully symmetric spaces of measurable functions with non-trivial Boyd indices is studied. In particular, it is shown that for such spaces Bourgain's Return Times theorem is valid.

math.FA↗
arXiv2013

Good Modulating Sequences for the Ergodic Hilbert Transform

This article investigates classes of bounded sequences of complex numbers that are universally good for the ergodic Hilbert transform in L_p-spaces, 2\leq p\leq \infty : The class of bounded Besicovitch sequences satisfying a rate condition is among such sequence classes.

math.DS↗