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Alex Stokolos

Publications and source records attributed to Alex Stokolos.

16 recordsLinked to original sources
arXiv2024

Remarks on the construction of $K_\sigma$ sets associated to trees not satisfying a separation condition

$K_\sigma$ sets involving sticky maps $\sigma$ have been used in the theory of differentiation of integrals to probabilistically construct Kakeya-type sets that imply certain types of directional maximal operators are unbounded on $L^p(\mathbb{R}^2)$ for all $1 \leq p < \infty$. We indicate limits to this approach by showing that, given $\epsilon > 0$ and a natural number $N$, there exists a tree $\mathcal{T}_{N, \epsilon}$ of finite height that is lacunary of order $N$ but such that, for \emph{every} sticky map $\sigma: \mathcal{B}^{h(\mathcal{T}_{N, \epsilon})} \rightarrow \mathcal{T}_{N, \epsilon}$, one has $|K_{\sigma} \cap ((1,2) \times \mathbb{R})| \geq 1 - \epsilon$.

math.CA↗
arXiv2022

$L^p(\mathbb{R}^2)$ bounds for geometric maximal operators associated to homothecy invariant convex bases

Let $\mathcal{B}$ be a nonempty homothecy invariant collection of convex sets of positive finite measure in $\mathbb{R}^2$. Let $M_\mathcal{B}$ be the geometric maximal operator defined by $$M_\mathcal{B}f(x) = \sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_R |f|\;.$$ We show that either $M_\mathcal{B}$ is bounded on $L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$ or that $M_\mathcal{B}$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. As a corollary, we have that any density basis that is a homothecy invariant collection of convex sets in $\mathbb{R}^2$ must differentiate $L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$.

math.CA↗
arXiv2022

Extremal problems for trinomials with fold symmetry

The famous T. Suffridge polynomials have many extremal properties: the maximality of coefficients when the leading coefficient is maximal; the zeros of the derivative are located on the unit circle; the maximum radius of stretching the unit disk with the schlicht normalization $F(0)=0$, $F'(0)=1$; the maximum size of the unit disk contraction in the direction of the real axis for univalent polynomials with the normalization $F(0)=0$, $F(1)=1.$ However, under the standard symmetrization method $\sqrt[T]{F(z^T)}$, these polynomials go to functions, which are not polynomials. How can we construct the polynomials with fold symmetry that have properties similar to those of the Suffridge polynomial? What values will the corresponding extremal quantities take in the above-mentioned extremal problems? The paper is devoted to solving these questions for the case of the trinomials $F(z)=z+az^{1+T}+bz^{1+2T}$. Also, there are suggested hypotheses for the general case in the work.

math.CV↗
arXiv2022

Sharp Weak Type Estimates for Maximal Operators associated to Rare Bases

Let $\mathcal{B}$ denote a nonempty translation invariant collection of intervals in $\mathbb{R}^n$ (which we regard as a rare basis), and define the associated geometric maximal operator $M_\mathcal{B}$ by $$M_\mathcal{B}f(x) = \sup_{x \in R \in \mathcal{B}} \frac{1}{|R|}\int_R |f|.$$ We provide a sufficient condition on $\mathcal{B}$ so that the estimate $$ |\{x \in \mathbb{R}^n : M_{\mathcal{B}}f(x) > α\}|\leq C_n \int_{\mathbb{R}^{n}} \frac{|f|}α\left(1+\log^+\frac{|f|}α\right)^{n-1} $$ is sharp. As a corollary we obtain sharp weak type estimates for maximal operators associated to several classes of rare bases including Córdoba, Soria and Zygmund bases.

math.CA↗
arXiv2021

Sharp Weak Type Estimates for a Family of Zygmund Bases

Let $\mathcal{B}$ be a collection of rectangular parallelepipeds in $\mathbb{R}^3$ whose sides are parallel to the coordinate axes and such that $\mathcal{B}$ consists of parallelepipeds with side lengths of the form $s, 2^j s, t $, where $s, t > 0$ and $j$ lies in a nonempty subset $S$ of the integers. In this paper, we prove the following: If $S$ is a finite set, then the associated geometric maximal operator $M_\mathcal{B}$ satisfies the weak type estimate of the form $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} \frac{|f|}α\left(1 + \log^+ \frac{|f|}α\right)\;$$ but does not satisfy an estimate of the form $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} ϕ\left(\frac{|f|}α\right)$$ for any convex increasing function $ϕ: \mathbb[0, \infty) \rightarrow [0, \infty)$ satisfying the condition $$\lim_{x \rightarrow \infty}\frac{ϕ(x)}{x (\log(1 + x))} = 0\;.$$ On the other hand, if $S$ is an infinite set, then the associated geometric maximal operator $M_\mathcal{B}$ satisfies the weak type estimate $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} \frac{|f|}α \left(1 + \log^+ \frac{|f|}α\right)^{2}$$ but does not satisfy an estimate of the form $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} ϕ\left(\frac{|f|}α\right)$$ for any convex increasing function $ϕ: \mathbb[0, \infty) \rightarrow [0, \infty)$ satisfying the condition $$\lim_{x \rightarrow \infty}\frac{ϕ(x)}{x (\log(1 + x))^2} = 0\;.$$

math.AP↗
arXiv2021

Univalent polynomials and Koebe's one-quarter theorem

The famous Koebe $\frac14$ theorem deals with univalent (i.e., injective) analytic functions $f$ on the unit disk $\mathbb D$. It states that if $f$ is normalized so that $f(0)=0$ and $f'(0)=1$, then the image $f(\mathbb D)$ contains the disk of radius $\frac14$ about the origin, the value $\frac14$ being best possible. Now suppose $f$ is only allowed to range over the univalent polynomials of some fixed degree. What is the optimal radius in the Koebe-type theorem that arises? And for which polynomials is it attained? A plausible conjecture is stated, and the case of small degrees is settled.

math.CV↗
arXiv2021

Sharp weak type estimates for a family of Soria bases

Let $\mathcal{B}$ be a collection of rectangular parallelepipeds in $\mathbb{R}^3$ whose sides are parallel to the coordinate axes and such that $\mathcal{B}$ contains parallelepipeds with side lengths of the form $s, \frac{2^N}{s} , t $, where $s, t > 0$ and $N$ lies in a nonempty subset $S$ of the natural numbers. We show that if $S$ is an infinite set, then the associated geometric maximal operator $M_\mathcal{B}$ satisfies the weak type estimate $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} \frac{|f|}α \left(1 + \log^+ \frac{|f|}α\right)^{2}$$ but does not satisfy an estimate of the form $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} ϕ\left(\frac{|f|}α\right)$$ for any convex increasing function $ϕ: \mathbb[0, \infty) \rightarrow [0, \infty)$ satisfying the condition $$\lim_{x \rightarrow \infty}\frac{ϕ(x)}{x (\log(1 + x))^2} = 0\;.$$

math.CA↗
arXiv2020

On Suffridge polynomials

We consider some known and some new properties of the family of polynomials introduced by Ted Suffridge in 1969. We begin by giving a brief overview of their extremal properties in classic and more recent work. We also give a compact form for Suffridge polynomials which matches a general pattern discovered by Brandt. Our approach allows us to find the coefficients which Brandt's result was not giving explicitly. This new presentation provides us the tools to obtain an estimate of the rate of approximation of the generalized Koebe functions by univalent polynomials. Furthermore, we consider the presentation of Suffridge polynomials in Robertson's form and find the suiting Robertson measure. This suggests a new way to approximate step functions by continuous monotonic ones. We then study the lack of robustness of the univalency of these polynomials and suggest a new family of polynomials for which we conjecture the univalency of a subclass. Namely, we prove the quite surprising fact that by extending the family by letting the discrete argument in the polynomial coefficients become continuous one does not increase the set of univalent polynomials. Only the initial polynomials remain univalent. In this new one parameter family generalizing the Suffridge polynomials, it is remarkable that the Suffridge polynomials are already extremal as they correspond to the choice of the parameter set to 1; moreover the complex Fejér polynomials correspond the choice of the parameter set to 0, and the choice of the parameter set to -1 corresponds the polynomials $z+(z^N/N)$. Remarkably, computer simulations seem to clearly indicate that the image of the unit disc under these new polynomial mapping is a simply-connected region bounded by a simple curve. This justifies the conjectural univalency of these polynomials for the whole range of the parameters.

math.CV↗
arXiv2020

On C. Michel's hypothesis about the modulus of typically real polynomials

Extremal problems for typically real polynomials go back to a paper by W. W. Rogosinski and G. Szegő, where a number of problems were posed, which were partially solved by using orthogonal polynomials. Since then, not too many new results on extremal properties of typically real polynomials have been obtained. Fundamental work in this direction is due to M.~Brandt, who found a novel way of solving extremal problems. In particular, he solved C. Michel's problem of estimating the modulus of a typically real polynomial of odd degree. On the other hand, D. K. Dimitrov showed the effectivity of Fejér's method for solving the Rogosinski--Szegő problems. In this article, we completely solve Michel's problem by using Fejér's method.

math.CA↗
arXiv2019

On the Koebe Quarter Theorem for Polynomials

D. Dimitrov has posed the problem of finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials and asked whether Suffridge polynomials are optimal. We disprove Dimitrov's conjecture for polynomials of degree 3, 4, 5 and 6. For polynomials of degree 1 and 2 the conjecture is obviously true. On the way we introduce a new family of polynomials that allows us to state a conjecture about the value of the Koebe radius for polynomials of a specific degree.

math.CV↗
arXiv2018

Dimitrov's question for the polynomials of degree 1,2,3,4,5,6

In 2002 Dimitar Dimitrov posted the problem of finding the optimal polynomials that provide the sharpness of Koebe Quarter Theorem for polynomials and asked whether Suffridge polynomials are optimal ones. We disproved Dimitrov's conjecture for polynomials of degree 3,4,5 and 6. For polynomials of degree 1 and 2 the conjecture is valid.

math.CV↗
arXiv2018

Estimating the Koebe radius for polynomials

For a pair of conjugate trigonometrical polynomials $C (t) = \sum_ { j = 1 } ^N { { a_j}\cos jt }, S(t) = \sum_ { j = 1 } ^N { { a_j}\sin jt }$ with real coefficients and normalization ${a_1} = 1 $ we solve the extremal problem \[ \sup_ {a_2,...,a_N} \left ({ \min_t \left\{ {\Re \left ({ F\left ({ { e^ {it} } } \right) } \right): \Im \left ({ F\left ({ { e^ {it} } } \right) } \right) = 0 } \right\} } \right) = -\frac14 \sec ^2\fracπ{N + 2}. \] We show that the solution is unique and is given by \[ a_j^ {(0)} = \frac {1} { { { U'_N}\left ({\cos \frac{π} { { N + 2 } } } \right) } } { U' _ { N - j + 1 } }\left ({\cos \frac{π} { { N + 2 } } } \right) { U_ { j - 1 } }\left ({\cos \frac{π} { { N + 2 } } } \right), \] where the $U_j(x)$ are the Chebyshev polynomials of the second kind, and the $U'_j(x)$ are their derivatives, $j = 1, \ldots, N.$ As a consequence, we obtain some theorems on covering of intervals by polynomial images of the unit disc. We formulate several conjectures on a number of extremal problems on classes of polynomials.

math.CV↗
arXiv2018

Fejer Polynomials and Control of Nonlinear Discrete Systems

We consider optimization problems associated to a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing $T$-cycles of a differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ of the form $$x(k+1) = f(x(k)) + u(k)$$ where $$u(k) = (a_1 - 1)f(x(k)) + a_2 f(x(k-T)) + \cdots + a_N f(x(k-(N-1)T))\;,$$ with $a_1 + \cdots + a_N = 1$. Following an approach of Morgül, we associate to each periodic orbit of $f$, $N \in \mathbb{N}$, and $a_1,\ldots,a_N$ an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. We prove that, given any 1- or 2-cycle of $f$, there exist $N$ and $a_1,\ldots,a_N$ whose associated polynomial is Schur stable, and we find the minimal $N$ that guarantees this stabilization. The techniques of proof will take advantage of extremal properties of the Fejér kernels found in classical harmonic analysis.

math.DS↗
arXiv2015

On the generalized linear and non-linear DFC in non-linear dynamics

The article is devoted to investigation of robust stability of the generalized linear control of the discrete autonomous dynamical systems. Sharp necessary conditions on the size of the set of multipliers that guaranty robust stabilization of the equilibrium of the system are provided. Surprisingly enough it turns out that the generalized linear delayed feedback control has same limitation as the classical Pyragas DFC. This generalized Ushio 1996 DFC limitation statement. Note that in scalar case a generalized non-linear control can robustly stabilize an equilibrium for any admissible range of multipliers. In the current article similar result is obtained in the vector-valued setting.

math.DS↗
arXiv2014

Fejer and Suffridge polynomials in the delayed feedback control theory

A remarkable connection between optimal delayed feedback control (DFC) and complex polynomial mappings of the unit disc is established. The explicit form of extremal polynomials turns out to be related with the Fejer polynomials. The constructed DFC can be used to stabilize cycles of one-dimensional non-linear discrete systems.

math.DS↗