arXiv subjects

Habibulla Akhadkulov

Publications and source records attributed to Habibulla Akhadkulov.

5 recordsLinked to original sources

arXiv2020

Equivalence of Weighted DT-Moduli of (Co)convex Functions

The paper present new definitions for weighted DT moduli. Similarly, we a general outcome in an equivalence of moduli of smoothness are obtained. It is known that, any $r \in \mathbb{N}_{\circ}$ , $0<p \leq \infty$, $1 \leq η\leq r$ and $ϕ(x)=\sqrt{1-x^2}$, the inequalities $ω^ϕ_{i+1,r} \; (f^{(r)}, \| θ_{\mathcal{N}} \|)_{w_{α, β}, p} \sim ω^ϕ_{i,r+1} \; (f^{(r+1)}, \| θ_{\mathcal{N}} \|)_{w_{α, β}, p}$ and $ω^ϕ_{i+η} \; (f, \| θ_{\mathcal{N}} \|)_{α, β, p} \sim \| θ_{\mathcal{N}} \|^{- η} ω^ϕ_{i, 2 η} \; (f^{(2 η)}, \| θ_{\mathcal{N}} \|)_{α+ η, β+ η, p}$ are valid.

math.FA↗

arXiv2020

Derivation of new Degrees for Best (Co)convex and Unconstrained Polynomial Approximation in $\mathbb{L}^{α, β}_p$ space: I

The purpose of this work is to present the derivation and an estimate of the degrees of the best approximation based on convex, coconvex and unconstrained polynomials, and discuss some applications. We simplify the term convex and coconvex polynomial as (co)convex polynomial herein.

math.FA↗

arXiv2016

Renormalization of circle diffeomorphisms with a break-type singularity

Let $f$ be an orientation-preserving circle diffeomorphism with irrational rotation number and with a break point $ξ_{0},$ that is, its derivative $f'$ has a jump discontinuity at this point. Suppose that $f'$ satisfies a certain Zygmund condition dependent on a parameter $γ>0.$ We prove that the renormalizations of $f$ are approximated by Möbius transformations in $C^{1}$-norm if $γ\in (0,1]$ and they are approximated in $C^{2}$-norm if $γ\in (1,+\infty).$ It is also shown, that the coefficients of Möbius transformations get asymptotically linearly dependent.

math.DS↗

arXiv2013

Nonrigidity of piecewise-smooth circle maps

Let $f_{i},$ $i=1,2$ be piecewise-smooth $C^{1}$ circle homeomorphisms with two break points, $\log Df_{i},$ $i=1,2$ are absolutely continuous on each continuity intervals of $Df_{i}$ and $D\log Df_{i}\in L^{p}$ for some $p>1.$ Suppose, the jump ratios of $f_{1} $ and $f_{2} $ at their break points do not coincide but have the same total jumps (i.e. the product of jump ratios) and identical irrational rotation number of bounded type. Then the conjugation $h$ between $f_{1} $ and $f_{2} $ is a singular function, i.e. it is continuous on $S^1,$ but $Dh(x)=0$ a.e. with respect to Lebesgue measure.

math.DS↗

arXiv2012

On conjugations of circle homeomorphisms with two break points

Let $f_i\in C^{2+α}(S^1\setminus \{a_i,b_i\}), α>0, i=1,2$ be circle homeomorphisms with two break points $a_i,b_i$, i.e. discontinuities in the derivative $f_i$, with identical irrational rotation number $rho$ and $μ_1([a_1,b_1])= μ_2([a_2,b_2])$, where $μ_i$ are invariant measures of $f_i$. Suppose the products of the jump ratios of $Df_1$ and $Df_2$ do not coincide, i.e. $\frac{Df_1(a_1-0)}{Df_1(a_1+0)}\times \frac{Df_1(b_1-0)}{Df_1(b_1+0)}\neq \frac{Df_2(a_2-0)}{Df_2(a_2+0)}\times \frac{Df_2(b_2-0)}{Df_2(b_2+0)}$. Then the map $ψ$ conjugating $f_1$ and $f_2$ is a singular function, i.e. it is continuous on $S^1$, but $Dψ= 0$ a.e. with respect to Lebesgue measure

math.DS↗