2018
Том 70
№ 4

# Kofanov V. A.

Articles: 24
Article (Russian)

### Sharp Remez-type inequalities of various metrics in the classes of functions with а given comparison function

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1472-1485

For any $p \in [1,\infty ],\; \omega > 0, \;\beta \in (0, 2\omega )$, and any measurable set $B \subset I_d := [0, d], \mu B \leq \beta$, we obtain the following sharp Remez-type inequality of various metrics $$E_0(x)\infty \leq \frac{\| \varphi \|_{\infty} }{E_0 (\varphi )L_p(I_{2\omega} \setminus B_1)}\| x\|_{ L_p(I_d\setminus B)}$$ on the classes $S_{\varphi} (\omega )$ of $d$-periodic $(d \geq 2\omega)$ functions $x$ with a given sine-shaped $2\omega$ -periodic comparison function $\varphi$, where $B_1 := [(\omega \beta )/2, (\omega + \beta )/2], E_0(f)L_p(G)$ is the best approximation of the function $f$ by constants in the metric of the space $L_p(G)$. In particular, we prove sharp Remez-type inequalities of various metrics in the Sobolev spaces of differentiable periodic functions. We also obtain inequalities of this type in the spaces of trigonometric polynomials and splines.

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)

### Sharp Remez-type inequalities for differentiable periodic functions, polynomials and splines

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 227-240

For any $\omega > 0,\; \beta \in (0, 2\omega)$, and any measurable set $B \in I_d := [0, d],\; \mu B = \beta$, we obtain the following sharp inequality of the Remez type: $$||x||_{\infty} \leq \frac{3||\varphi||_{\infty} - \varphi \biggl(\frac{\omega - \beta}2 \biggr)}{||\varphi||_{\infty} + \varphi \biggl(\frac{\omega - \beta}2 \biggr)} ||x||_{L_{\infty}(I_d\setminus B)}$$ on the set $S_{\varphi} (\omega )$ of functions $x$ with minimal period $d (d \geq 2\omega)$ and a given sine-shaped $2\omega$ -periodic comparison function $\varphi$. In particular, we prove the sharp Remez-type inequalities on the Sobolev spaces of differentiable periodic functions. We also obtain inequalities of the indicated type on the spaces of trigonometric polynomials and polynomial splines.

Anniversaries (Ukrainian)

### Motornyi Vitalii Pavlovych (on his 75th birthday)

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 995-999

Article (Russian)

### Inequalities of Different Metrics for Differentiable Periodic Functions

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 202–212

We prove the following sharp inequality of different metrics: $$\begin{array}{cc}\hfill {\left\Vert x\right\Vert}_q\le {\left\Vert {\varphi}_r\right\Vert}_q{\left(\frac{{\left\Vert x\right\Vert}_p}{{\left\Vert {\varphi}_r\right\Vert}_p}\right)}^{\frac{r+1/q}{r+1/p}}{\left\Vert {x}^{(r)}\right\Vert}_{\infty}^{\frac{1/p-1/q}{r+1/p}},\hfill & \hfill q>p>0,\hfill \end{array}$$ for 2π -periodic functions $x ∈ L_{∞}^r$ satisfying the condition $$L{(x)}_p\le {2}^{1/p}{\left\Vert x\right\Vert}_p,$$ where $$L{(x)}_p:= \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:a,b\in \left[0,2\pi \right],\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\},$$ and $φ_r$ is the Euler spline of order $r$. As a special case, we establish the Nikol’skii-type sharp inequalities for polynomials and polynomial splines satisfying the condition (A).

Article (Ukrainian)

### Inequalities for Nonperiodic Splines on the Real Axis and Their Derivatives

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 216–225

We solve the following extremal problems: (i) ${\left\Vert {s}^{(k)}\right\Vert}_{L_q\left[\alpha, \beta \right]}\to \sup$ and (ii) ${\left\Vert {s}^{(k)}\right\Vert}_{W_q}\to \sup$ over all shifts of splines of order r with minimal defect and nodes at the points lh, l ∈ Z , such that L(s) p ≤M in the cases: (a) k =0, q ≥ p >0, (b) k =1, . . . , r −1, q ≥ 1, where [α, β] is an arbitrary interval in the real line, $$L{(x)}_p:= \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:a,b\in \mathbf{R},\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\}$$

and ${\left\Vert \cdot \right\Vert}_{W_q}$ is the Weyl functional, i.e., $${\left\Vert x\right\Vert}_{W_q}:=\underset{\varDelta \to \infty }{ \lim}\underset{a\in \mathbf{R}}{ \sup }{\left(\frac{1}{\varDelta }{\displaystyle \underset{a}{\overset{a+\varDelta }{\int }}{\left|x(t)\right|}^qdt}\right)}^{1/q}.$$

As a special case, we get some generalizations of the Ligun inequality for splines.

Article (Russian)

### Inequalities for derivatives of functions on an axis with nonsymmetrically bounded higher derivatives

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 636-648

For nonperiodic functions $x \in L^r_{\infty}(\textbf{R})$ defined on the entire real axis, we prove analogs of the Babenko inequality. The obtained inequalities estimate the norms of derivatives $||x^{(k)}_{\pm}||_{L_q[a, b]}$ on an arbitrary interval $[a,b] \subset R$ such that $x^{(k)}(a) = x^{(k)}(b) = 0$ via local $L_p$-norms of the functions $x$ and uniform nonsymmetric norms of the higher derivatives $x(r)$ of these functions.

Article (Russian)

### Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 969-984

For arbitrary $[\alpha, \beta] \subset \textbf{R}$ and $p > 0$, we solve the extremal problem $$\int_{\alpha}^{\beta}|x^{(k)}(t)|^q dt \rightarrow \sup, \quad q \geq p, \quad k = 0, \quad \text{or} \quad q \geq 1, \quad k \geq 1,$$ on the set of functions $S^k_{\varphi}$ such that$\varphi ^{(i)}$ is the comparison function for $x^{(i)},\; i = 0, 1, . . . , k$, and (in the case $k = 0$) $L(x)_p \leq L(\varphi)_p$, where $$L(x)_p := \sup \left\{\left(\int^b_a|x(t)|^p dt \right)^{1/p}\; :\; a, b \in \textbf{R},\; |x(t)| > 0,\; t \in (a, b) \right\}$$ In particular, we solve this extremal problem for Sobolev classes and for bounded sets of the spaces of trigonometric polynomials and splines.

Article (Russian)

### On some extremal problems of different metrics for differentiable functions on the axis

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 765-776

For an arbitrary fixed segment $[α, β] ⊂ R$ and given $r ∈ N, A_r, A_0$, and $p > 0$, we solve the extremal problem $$∫^{β}_{α} \left|x^{(k)}(t)\right|^qdt → \sup,\; q⩾p,\; k=0,\; q⩾1,\; 1 ⩽ k ⩽ r−1,$$ on the set of all functions $x ∈ L^r_{∞}$ such that $∥x (r)∥_{∞} ≤ A_r$ and $L(x)_p ≤ A_0$, where $$L(x)p := \left\{\left( ∫^b_a |x(t)|^p dt\right)^{1/ p} : a,b ∈ R,\; |x(t)| > 0,\; t ∈ (a,b)\right\}$$ In the case where $p = ∞$ and $k ≥ 1$, this problem was solved earlier by Bojanov and Naidenov.

Article (Russian)

### On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1642–1649

New sharp inequalities of the Kolmogorov type are established, in particular, the following sharp inequality for $2 \pi$-periodic functions $x \in L^r_{\infty}(T):$ $$||x^{(k)}||_l \leq \left(\frac{\nu(x')}{2} \right)^{\left(1 - \frac1p \right)\alpha} \frac{||\varphi_{r-k}||_l}{||\varphi_r||^{\alpha}_p} ||x||^{\alpha}_p \left|\left|x^{(r)}\right|\right|^{1-\alpha}_{\infty},$$ $k,\;r \in \mathbb{N},\quad k < r, \quad r \geq 3,\quad p \in [1, \infty],\quad \alpha = (r-k) / (r - 1 + 1/p), \quad \varphi_r$ is the perfect Euler spline of order $r,\quad \nu(x')$ is the number of sign changes of the derivative $x'$ on a period.

Article (Russian)

### Inequalities for derivatives of functions in the spaces Lp

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1338 – 1349

The following sharp inequality for local norms of functions $x \in L^{r}_{\infty,\infty}(\textbf{R})$ is proved: $$\frac1{b-a}\int\limits_a^b|x'(t)|^qdt \leq \frac1{\pi}\int\limits_0^{\pi}|\varphi_{r-1}(t)|^q dt \left(\frac{||x||_{L_{\infty}(\textbf{R})}}{||\varphi_r||_{\infty}}\right)^{\frac{r-1}rq}||x^{(r)}||^q_{\infty}r,\quad r \in \textbf{N},$$ where $\varphi_r$ is the perfect Euler spline, takes place on intervals $[a, b]$ of monotonicity of the function $x$ for $q \geq 1$ or for any $q > 0$ in the cases of $r = 2$ and $r = 3.$ As a corollary, well-known A. A. Ligun's inequality for functions $x \in L^{r}_{\infty}$ of the form $$||x^{(k)}||_q \leq \frac{||\varphi_{r-k}||_q}{||\varphi_r||_{\infty}^{1-k/r}} ||x||^{1-k/r}_{\infty}||x^{(r)}||^{k/r}_{\infty},\quad k,r \in \textbf{N},\quad k < r, \quad 1 \leq q < \infty,$$ is proved for $q \in [0,1)$ in the cases of $r = 2$ and $r = 3.$

Article (Russian)

### On exact Bernstein-type inequalities for splines

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1357–1367

We establish new exact Bernstein-type and Kolmogorov-type inequalities. The main result of this work is the following exact inequality for periodic splines $s$ of order $r$ and defect 1 with nodes at the points $iπ/n, i ∈ Z, n ∈ N:$ $$\left\| {s^{(k)} } \right\|_q \leqslant n^{k + 1/p - 1/q} \frac{{\left\| {\varphi _{r - k} } \right\|_q }}{{\left\| {\varphi _r } \right\|_p }}\left\| s \right\|_p ,$$ where $k, r ∈ N, k < r, p = 1$ or $p = 2, q > p$, and $ϕr$ is the perfect Euler spline of order $r$.

Article (Russian)

### Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 291–302

We obtain new exact inequalities of the form $$∥x(k)∥_q ⩽ K∥x∥^{α}_p ∥x(r)∥^{1−α}_s$$ for functions defined on the axis $R$ or the semiaxis $R_{+}$ in the case where $$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s=1,$$ for functions defined on the axis $R$ in the case where $$r = 2,\; k = 1,\; q ∈ [2,∞),\; p = ∞,\; s= 1,$$ and for functions of constant sign on $R$ or $R_{+}$ in the case where $$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s = ∞$$ and in the case where $$r = 2,\; k = 1,\; p ∈ (0,∞),\; q = s = ∞.$$

Article (Russian)

### On the set of extremal functions in certain Kolmogorov-type inequalities

Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1062–1075

We determine the sets of all extremal functions in certain Kolmogorov-type and Bohr-Favard-type inequalities.

Article (Russian)

### Approximation of sine-shaped functions by constants in the spaces $L_p,\; p < 1$

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 745–762

We investigate the best approximations of sine-shaped functions by constants in the spaces $L_p$ for $p < 1$. In particular, we find the best approximation of perfect Euler splines by constants in the spaces Lp for certain $p∈(0,1)$.

Article (Russian)

### Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 579-589

We investigate the relationship between the constants K(R) and K(T), where $K\left( G \right) = K_{k,r} \left( {G;q,p,s;\alpha } \right): = \mathop {\mathop {\sup }\limits_{x \in L_{p,s}^r \left( G \right)} }\limits_{x^{(r)} \ne 0} \frac{{\left\| {x^{\left( k \right)} } \right\|_{L_q \left( G \right)} }}{{\left\| x \right\|_{L_q \left( G \right)}^\alpha \left\| {x^{\left( r \right)} } \right\|_{L_s \left( G \right)}^{1 - \alpha } }}$ is the exact constant in the Kolmogorov inequality, R is the real axis, T is a unit circle, $$L_{p,s}^r (G)$$ is the set of functions xL p(G) such that x (r)L s(G), q, p, s ∈ [1, ∞], k, rN, k < r, We prove that if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} = 1 - k/r$$ thenK(R) = K(T),but if $$\frac{r - k + 1/q - 1/s}{r + 1/q - 1/s} < 1 - k/r$$ thenK(R) ≤ K(T); moreover, the last inequality can be an equality as well as a strict inequality. As a corollary, we obtain new exact Kolmogorov-type inequalities on the real axis.

Article (Russian)

### On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 456-469

For 2π-periodic functions $x \in L_\infty ^r$ and arbitrary q ∈ [1, ∞] and p ∈ (0, ∞], we obtain the new exact Kolmogorov-type inequality $|| x^(k) ||_q \leqslant (\frac{v(x^(k))}{2})^{1/q} \frac{|| \phi_{r-k} ||_q}{||| \phi_r |||_p^\alpha} ||| x |||_p^\alpha || x^(r) ||_\infty^{1- \alpha}, k, r \in N, k < r,$ which takes into account the number of changes in the sign of the derivatives ν(x (k)) over the period. Here, α = (rk + 1/q)/(r + 1/p), ϕ r is the Euler perfect spline of degree r, $\begin{gathered} \left\| {\left| x \right|} \right\|_p : = {\text{sup}}_{a,b \in {\text{R}}} \{ E_0 (x)_{L_p [a,b]} :x'(t) \ne 0{\text{ }}\forall t \in (a,b)\} , \\ {\text{ }} \\ {\text{ }}E_0 (x)_{L_p [a,b]} : = {\text{ inf}}_{c \in {\text{R}}} \left\| {x - c} \right\|_{L_p [a,b]} , \\ \\ \left\| x \right\|_{L_p [a,b]} : = \left\{ {\int\limits_a^b {\left| {x(t)} \right|^p dt} } \right\}^{1/p} {\text{ for }}0 < p < \infty , \\ \end{gathered}$ and $\left\| x \right\|_{L_p [a,b]} : = {\text{ sup vrai}}_{t \in \left[ {a,b} \right]} \left| {x(t)} \right|$ . The inequality indicated turns into the equality for functions of the form x(t) = aϕ r (nt + b), a, bR, nN. We also obtain an analog of this inequality in the case where k = 0 and q = ∞ and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines.

Brief Communications (Russian)

### On Kolmogorov-Type Inequalities with Integrable Highest Derivative

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1694-1697

We obtain the new exact Kolmogorov-type inequality $$\left\| {x^{\left( k \right)} } \right\|_2 \leqslant K\left\| x \right\|_2^{\frac{{r - k - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}{{r - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \left\| {x^{\left( r \right)} } \right\|_1^{\frac{k}{{r{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}$$ for 2π-periodic functions $x \in L_1^r$ and any k, rN, k < r. We present applications of this inequality to problems of approximation of one class of functions by another class and estimates of K-functional type.

Article (Russian)

### Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications

Ukr. Mat. Zh. - 2002. - 54, № 10. - pp. 1348-1356

We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions xL x (r), namely, $$\left\| {x^{(k)} } \right\|_{L_\infty (R)} \leqslant \frac{{\left\| {\phi _{r - k} } \right\|_\infty }}{{\left\| {\phi _r } \right\|_\infty ^{1 - k/r} }}M(x)^{1 - k/r} \left\| {x^{(r)} } \right\|_{L_\infty (R)}^{k/r} ,$$ where $$M(x): = \frac{1}{2}\mathop {\sup }\limits_{\alpha ,\beta } \left\{ {\left| {x(\beta ) - x(\alpha )} \right|:x'(t) \ne 0{\text{ }}\forall t \in (\alpha ,\beta )} \right\}{\text{,}}$$ k, rN, k < r, and ϕ r is a perfect Euler spline of order r. Using this inequality, we strengthen the Bernstein inequality for trigonometric polynomials and the Tikhomirov inequality for splines. Some other applications of this inequality are also given.

Article (Russian)

### Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 603-609

We obtain a new unimprovable Kolmogorov-type inequality for differentiable 2π-periodic functions x with bounded variation of the derivative x′, namely $$\left\| {x'} \right\|_q \leqslant K\left( {q,p} \right)\left\| x \right\|_p^a \left( {\mathop V\limits_{0}^{{2\pi }} \left( {x'} \right)} \right)^{1 - {alpha }} ,$$ where q ∈ (0, ∞), p ∈ [1, ∞], and α = min{1/2, p/q(p + 1)}.

Article (Russian)

### Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1299-1308

We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p ∈ [1, ∞], the following unimprovable inequality holds for functions $x \in L_\infty ^r$ : $$\left\| {x^{\left( k \right)} } \right\|_q \leqslant \frac{{\left\| {{\phi }_{r - k} } \right\|_q }}{{\left\| {{\phi }_r } \right\|_p^\alpha }}\left\| x \right\|_p^\alpha \left\| {x^{\left( k \right)} } \right\|_\infty ^{1 - \alpha }$$ where $\alpha = \min \left\{ {1 - \frac{k}{r},\frac{{r - k + {1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-0em} q}}}{{r + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0em} p}}}} \right\}$ and ϕ r is the perfect Euler spline of order r.

Article (Ukrainian)

### Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 597-609

We obtain new inequalities of different metrics for differentiable periodic functions. In particular, for p, q ∈ (0, ∞], q > p, and s ∈ [p, q], we prove that functions $x \in L_\infty ^{{\text{ }}r}$ satisfy the unimprovable inequality $$|| (x-c_{s+1} (x))_{\pm} ||_q \leqslant \frac{|| (\phi_r)_{\pm} ||_q}{|| \phi_r ||_p^{\frac{r+1/q}{r+1/p}}} || x-c_{s+1}(x) ||_p^{\frac{r+1/q}{r+1/P}} || x^(r) ||_\infty^{\frac{1/p-1/q}{r+1/p}},$$ where ϕ r is the perfect Euler spline of order r and c s + 1(x) is the constant of the best approximation of the function x in the space L s + 1. By using the inequality indicated, we obtain a new Bernstein-type inequality for trigonometric polynomials τ whose degree does not exceed n, namely, $$|| (\tau^(k))_{\pm} ||_q \leqslant n^{k+1/p-1/q} \frac{|| (\cos(\cdot))_{\pm} ||_q}{|| \cos(\cdot) ||_p} || \tau ||_p,$$ where kN, p ∈ (0, 1], and q ∈ [1, ∞]. We also consider other applications of the inequality indicated.

Article (Russian)

### On additive inequalities for intermediate derivatives of functions given on a finite interval

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 619–628

We present a general scheme for deducing additive inequalities of Landau-Hadamard type. As a consequence, we prove several new inequalities for the norms of intermediate derivatives of functions given on a finite interval with an exact constant with the norm of a function.

Article (Ukrainian)

### Massiveness of the sets of extremal functions in some problems in approximation theory

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1356–1361

It is proved that the sets of extremal functions are massive in some problems in approximation theory.