2017
Том 69
№ 7

All Issues

Pichugov S. A.

Articles: 25
Brief Communications (Russian)

Nikol’skii – Stechkin-type inequalities for the increments of trigonometric polynomials in metric spaces

Pichugov S. A.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 711-716

In the spaces $L_{\Psi} [0, 2\pi ]$ with the metric $$\rho (f, 0)\Psi = \frac1{2\pi }\int^{2\pi }_0 \Psi (| f(x)| ) dx,$$ where $\Psi$ is a function of the modulus-ofcontinuity type, we investigate an analog of the Nikol’skii – Stechkin inequalities for the increments and derivatives of trigonometric polynomials.

Article (Russian)

Some properties of the moduli of continuity of periodic functions in metric spaces

Pichugov S. A.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1657-1664

Let $L_0(T)$) be the set of real-valued periodic measurable functions, let $\Psi : R^{+} \rightarrow R^{+}$ be the modulus of continuity, and let $$L_{\Psi} \equiv L_{\Psi} (T) = \left\{ f \in L_0(T) : \| f\| _{\Psi} := \frac1{2\pi} \int_T \Psi (| f(x)| )dx < \infty \right\}.$$ We study the properties of multiple modules of continuity for the functions from $L_{\Psi}$.

Article (Russian)

Smoothness of functions in the metric spaces Lψ

Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1214-1232

Let $L_0(T)$ be thе set of real-valued periodic measurable functions, let $\psi : R^+ \rightarrow R^+$ be a modulus of continuity $(\psi \neq 0)$ , and let $$L_{\psi} \equiv L_{\psi}(T ) = \left\{f \in L_0 (T ): ||f||_{\psi} := \int_T \psi( |f (x)| ) dx < \infty \right\}.$$ The following problems are investigated: Relationship between the rate of approximation of $f$ by trigonometric polynomials in $L_{\psi}$ and smoothness in $L_1$. Correlation between the moduli of continuity of $f$ in $L_{\psi}$ and $L_1$, and theorems on imbedding of the classes $\text{Lip} (\alpha, \psi)$ in $L_1$. Structure of functions from the class $\text{Lip}(1, \psi)$.

Article (Russian)

Lower bounds for the deviations of the best linear methods of approximation of continuous functions by trigonometric polynomials

Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 662-673

In the case of uniform approximation of continuous periodic functions of one variable by trigonometric polynomials, we obtain lower bounds for the Jackson constants of the best linear methods of approximation.

Article (Russian)

Inverse Jackson theorems in spaces with integral metric

Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 351-362

In the spaces $L_{\Psi}(T)$ of periodic functions with metric $\rho(f, 0)_{\Psi} = \int_T \Psi(|f(x)|)dx$, where $\Psi$ is a function of the modulus-of-continuity type, we investigate the inverse Jackson theorems in the case of approximation by trigonometric polynomials. It is proved that the inverse Jackson theorem is true if and only if the lower dilation exponent of the function $\Psi$ is not equal to zero.

Article (Russian)

Inequalities for trigonometric polynomials in spaces with integral metric

Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1657-1671

In the spaces $L_{\psi}(T)$ of periodic functions with metric $\rho( f , 0)_{\psi} = \int_T \psi (| f (x) |) dx $, where $\psi$ is a function of the modulus-of-continuity type, we investigate analogs of the classic Bernstein inequalities for the norms of derivatives and increments of trigonometric polynomials.

Article (Russian)

On the Jackson theorem for periodic functions in metric spaces with integral metric. II

Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1524-1533

In the spaces $L_{\psi}(T^m)$ of periodic functions with metric $\rho(f, 0)_{\psi} = \int_{T^m}\psi(|f(x)|)dx$ , where $\psi$ is a function of the type of modulus of continuity, we study the direct Jackson theorem in the case of approximation by trigonometric polynomials. It is proved that the direct Jackson theorem is true if and only if the lower dilation index of the function $\psi$ is not equal to zero.

Article (English)

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Babenko V. F., Parfinovych N. V., Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 301–314

Let $C(\mathbb{R}^m)$ be the space of bounded and continuous functions $x: \mathbb{R}^m → \mathbb{R}$ equipped with the norm $∥x∥_C = ∥x∥_{C(\mathbb{R}^m)} := \sup \{ |x(t)|:\; t∈ \mathbb{R}^m\}$ and let $e_j,\; j = 1,…,m$, be a standard basis in $\mathbb{R}^m$. Given moduli of continuity $ω_j,\; j = 1,…, m$, denote $$H^{j,ω_j} := \left\{x ∈ C(\mathbb{R}^m): ∥x∥_{ω_j} = ∥x∥_{H^{j,ω_j}} = \sup_{t_j≠0} \frac{∥Δtjejx(⋅)∥_C}{ω_j(|t_j|)} < ∞\right\}.$$ We obtain new sharp Kolmogorov-type inequalities for the norms $∥D^{α}_{ε}x∥_C$ of mixed fractional derivatives of functions $x ∈ ∩^{m}_{j=1}H^{j,ω_j}$. Some applications of these inequalities are presented.

Brief Communications (Russian)

On Kolmogorov-type inequalities for fractional derivatives of functions of two variables

Babenko V. F., Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 837–842

We prove a new exact Kolmogorov-type inequality estimating the norm of a mixed fractional-order derivative (in Marchaud's sense) of a function of two variables via the norm of the function and the norms of its partial derivatives of the first order.

Article (Russian)

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

Babenko V. F., Kofanov V. A., Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

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)

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

Babenko V. F., Kofanov V. A., Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

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)

Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Babenko V. F., Korneichuk N. P., Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 579-594

Let $γ = (γ_1 ,..., γ_d )$ be a vector with positive components and let $D^γ$ be the corresponding mixed derivative (of order $γ_j$ with respect to the $j$ th variable). In the case where $d > 1$ and $0 < k < r$ are arbitrary, we prove that $$\sup_{x \in L^{r\gamma}_{\infty}(T^d)D^{r\gamma}x\neq0} \frac{||D^{k\gamma}x||_{L_{\infty}(T^d)}}{||x||^{1-k/r}||D^{r\gamma}||^{k/r}_{L_{\infty}(T^d)}} = \infty$$ and $$||D^{k\gamma}x||_{L_{\infty}(T^d)} \leq K||x||^{1 - k/r}_{L_{\infty}(T^d)}||D^{r\gamma}x||_{L_{\infty}(T^d)}^{k/r} \left(1 + \ln^{+}\frac{||D^{r\gamma}x||_{L_{\infty}(T^d)}}{||x||_{L_{\infty} (T^d)}}\right)^{\beta}$$ for all $x \in L^{r\gamma}_{\infty}(T^d)$ Moreover, if \(\bar \beta \) is the least possible value of the exponent β in this inequality, then $$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar \beta \left( {d,\gamma ,k,r} \right) \leqslant d - 1.$$

Article (Russian)

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

Babenko V. F., Kofanov V. A., Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

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.

Brief Communications (Russian)

On Kolmogorov-Type Inequalities with Integrable Highest Derivative

Babenko V. F., Kofanov V. A., Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

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)

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

Babenko V. F., Kofanov V. A., Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

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)

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

Babenko V. F., Kofanov V. A., Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

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)

Approximation of periodic functions by constants in the metric spaces ?p(L)

Pichugov S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1095–1098

By using the best approximations of functions by constants, we establish necessary conditions for the moduli of continuity of periodic functions in metric spaces with integral metric and find the Young constants of these spaces.

Article (Ukrainian)

Asymptotic behavior of the best approximations for the functions in Lp

Pichugov S. A.

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 867–870

Article (Ukrainian)

The Riesz formula for multiplier operators in a space of trigonometric polynomials

Pichugov S. A.

Full text (.pdf)

Ukr. Mat. Zh. - 1991. - 43, № 4. - pp. 453-455

Article (Ukrainian)

Jung's relative constant of the space Lp

Pichugov S. A.

Full text (.pdf)

Ukr. Mat. Zh. - 1990. - 42, № 1. - pp. 122–125

Article (Ukrainian)

Sharp estimates of approximation in Lp by functions of the form φ(x) + ψ(y)

Pichugov S. A.

Full text (.pdf)

Ukr. Mat. Zh. - 1989. - 41, № 6. - pp. 815-818

Article (Ukrainian)

Inequalities for the derivatives of polynomials with real zeros

Babenko V. F., Pichugov S. A.

Full text (.pdf)

Ukr. Mat. Zh. - 1986. - 38, № 4. - pp. 411–416

Article (Ukrainian)

Invalidity of the elements of best approximation and a theorem of Glaeser

Ganzburg M. I., Pichugov S. A.

Full text (.pdf)

Ukr. Mat. Zh. - 1981. - 33, № 5. - pp. 664—667

Article (Ukrainian)

A property of compact operators in the space of integrable functions

Babenko V. F., Pichugov S. A.

Full text (.pdf)

Ukr. Mat. Zh. - 1981. - 33, № 4. - pp. 491–492

Article (Ukrainian)

Approximation in the mean of linear combinations of shifts of certain functions

Babenko V. F., Pichugov S. A.

Full text (.pdf)

Ukr. Mat. Zh. - 1981. - 33, № 2. - pp. 234–240