2017
Том 69
№ 12

Volume 69, № 2, 2017

Article (Ukrainian)

Bernstein – Nikol’skii-type inequalities for trigonometric polynomials with an arbitrary choice of harmonics

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 147-156

We obtain the Bernstein – Nikol’skii-type inequalities for trigonometric polynomials with an arbitrary choice of harmonics.

Article (Russian)

Properties of strong random operators generated by an Arratia flow

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 157-172

We study the properties of strong random operators $T_t$ in $L_2(R)$ used to describe the shifts of the functions along an Arratia flow. We prove the formula of change of variables for the Arratia flow. As a consequence of this formula, we establish sufficient conditions for compact sets $K \subset L_2(R)$ under which $T_t$ has a continuous modification on $K$. We also present necessary and sufficient conditions for the convergent sequences in $L_2(R)$ under which the operator $T_t$ preserves their convergence.

Article (Russian)

Sharp Remez-type inequalities of various metrics for differentiable periodic functions, polynomials, and splines

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 173-188

We prove a sharp Remez-type inequality of various metrics $$\| x\| q \leq \| \varphi_r\| q \biggl\{\frac{\| x\|_{L_p([0,2\pi ]\setminus B)}}{\|\varphi r\|_{ L_p([0,2\pi ]\setminus B_1)}}\biggr\}^{\alpha } \| x(r)\|^{1 - \alpha}_{ \infty} ,\; q > p > 0, \;\alpha = (r + 1/q)/(r + 1/p),$$ for $2\pi$ -periodic functions $x \in L^r_{\infty}$ satisfying the condition $$L(x)p \leq 2^{-\frac 1p}\| x\|_p,\quad (\ast )$$ where $$L(x)p := \mathrm{s}\mathrm{u}\mathrm{p} \Bigl\{ \| x\| L_p[a,b] : [a, b] \subset [0, 2\pi ], | x(t)| > 0, t \in (a, b)\Bigr\},$$ $B \subset [0, 2\pi ], \mu B \leq \beta /\lambda$ ($\lambda$ is chosen so that $\| x\| p = \| \varphi \lambda ,r\| L_p[0,2\pi /\lambda ] ), \varphi_r$ is the ideal Euler’s spline of order r, and $$B_1 := \biggl[\frac{-\pi - \beta /2}{2} , \frac{-\pi + \beta /2}{2} \biggr] \bigcup \biggl[ \frac{\pi - \beta /2}{2}, \frac{\pi + \beta /2}{2} \biggr].$$ As a special case, we establish sharp Remez-type inequalities of various metrics for trigonometric polynomials and polynomial splines satisfying the condition $(\ast )$.

Article (Russian)

Estimates of the area of solutions of the pseudolinear differential equations with Hukuhara derivative in the space $\text{conv} (R^2)$

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 189-214

We obtain estimates for the areas of the solutions of differential equations with Hukuhara derivative of a special form in the space $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v} (R^2)$. The main methods used for the investigation are the method of comparison, the methods of the Minkowski – Aleksandrov geometry of convex bodies, and the Chaplygin –Wa˙zewski method of approximate integration of differential equations. The obtained results enable us to reduce the estimates of the area of solutions to the investigation of differential equations of the first order.

Article (Ukrainian)

Best one-sided approximation on the average for the classes of differentiable functions by algebraic polynomials

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 215-227

We establish the best asymptotic one-sided approximation on the average for $r$ -differentiable functions from the class $W_{∞}^r$ where r is even, by algebraic polynomials.

Article (English)

Jacobi operators and orthonormal matrix-valued polynomials. I

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 228-239

It is shown that every self-adjoint operator in a separable Hilbert space is unitarily equivalent to a block Jacobi operator. A system of orthogonal operator-valued polynomials is constructed.

Article (English)

Еxact rates in the Davis – Gut law of iterated logarithm for the first moment convergence of independent identically distributed random variables

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 240-256

Let $\{X, X_n, n \geq 1\}$ be a sequence of independent identically distributed random variables and let $S_n = \sum^n_{i=1} X_i$, $M_n = \max_{1\leq k\leq n} |S_k|$. For $r > 0$, let $a_n(\varepsilon)$ be a function of $\varepsilon$ such that $a_n(\varepsilon ) \mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} n \rightarrow \tau$ as $n \rightarrow \infty$ and $\varepsilon \searrow \surd r$. If $EX^2I\{|X| \geq t\} = o(\text{log}\text{log}t)^{-1})$ as $t \rightarrow \infty$ , then, by using the strong approximation, we show that $$\lim_{\varepsilon \searrow \surd r} \frac 1{-\text{log}(\varepsilon^2 - r)} \sum ^{\infty}_{n=1}\frac{(\text{log} n)^{r-1}}{n^{3/2}}E \Bigl\{ M_n - (\varepsilon + a_n(\varepsilon ))\sigma \sqrt{2n \text{log log} n} \Bigr\}_{+} = \frac{2\sigma \varepsilon^{-2\tau \sqrt{r}}}{\sqrt{2\pi}r}$$ holds if and only if $EX = 0, EX^2 = \sigma^2$, and $EX = 0, EX^2 = \sigma^2$ та $EX^2(\mathrm{l}\mathrm{o}\mathrm{g} | X| )^{r-1}(\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} | X| )^{-\frac 12} < \infty$.

Anniversaries (Ukrainian)

Oleksandr Mykolaiovych Sharkovs’kyi (on his 80th birthday)

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 257-260

Brief Communications (Ukrainian)

Estimates of the product of inner radii of five nonoverlapping domains

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 261-267

We study the extremal V. N. Dubinin problem in the geometric theory of functions of complex variables connected with the estimates of a functional defined on a system of nonoverlapping domains. A particular solution of this problem is obtained.

Brief Communications (English)

Remarks on a Bailey pair with one free parameter

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 268-272

We offer a more general Bailey pair than the pair obtained in the papers [Andrews G. E., Adv. Combin., Waterloo Workshop in Comput. Algebra (May 26 – 29, 2011), Springer (2013), p. 57 – 76] and [Patkowski A. E., Discrete Math., 310, 961 – 965 (2010)] by two different methods.

Brief Communications (Russian)

On the equicontinuity of mappings with branching in the closure of the domain

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 273-279