Alexander Babenko

Institute of mathematics and mechanics of Ural branch of the Russian academy of sciences , Russia, Yekaterinburg · Approximation and applications department

Brief information about S. B. Stechkin's workshop-conference on function theory for the fifty years of its existence is given. The 46th workshop-conference dedicated to the 85th anniversary of Corresponding Member of the Russian Academy of Sciences Yu. N. Subbotin and Honored Scientist of the Russian Federation N. I. Chernykh, which took place with...

For given \(k\in\mathbb{N}\) and \(h>0\), an exact inequality \(\|W_{2k}(f,h)\|_{C}\leq C_{k}\,\|f\|_{C}\) is considered on the space \(C=C(\mathbb{R})\) of continuous functions bounded on the real axis \(\mathbb{R}=(-\infty,\infty)\) for the Boman–Shapiro difference operator \(W_{2k}(f,h)(x):=\displaystyle\frac{(-1)^{k}}{h}\displaystyle\intop\noli...

For given $k\in\bfn$ and $h>0$, an exact inequality $\|W_{2k}(f,h)\|_{C}\le C_{k}\,\|f\|_{C}$ is considered on the space $C=C(\bfr)$ of continuous functions bounded on the real axis $\bfr=(-\infty,\infty)$ for the Boman--Shapiro difference operator $W_{2k}(f,h)(x):=\ds\frac{(-1)^k}{h}\ds\int\limits_{-h}^h\!{\binom {2k} k}^{\!-1}\widehat \Delta_t^{2...

P. L. Chebyshev posed and solved (1857, 1859) the problem of finding an improper rational fraction least deviating from zero in the uniform metric on a closed interval among rational fractions whose denominator is a fixed polynomial of a given degree m that is positive on the interval and whose numerator is a polynomial of a given degree n ≥ m with...

New estimates are proved for the constants J(k, α) in the classical Jackson–Stechkin inequality En−1(f) ≤ J(k, α)ωk(f,απ/n), α > 0, in the case of approximation of functions f ∈ C[−1, 1] by algebraic polynomials. The main result of the paper implies the following two-sided estimates for the constants: 1/2 ≤ J(2k, α) < 10, n ≥ 2k(2k − 1), α ≥ 2.

We consider the generalized Poisson kernel Πq,α = cos(απ/2)P + sin(απ/2)Q with q ∈ (−1, 1) and α ∈ ℝ, which is a linear combination of the Poisson kernel \(P(t) = 1/2 + \sum\nolimits_{k = 1}^\infty {{q^k}} \cos kt\) and the conjugate Poisson kernel \(Q(t) = \sum\nolimits_{k = 1}^\infty {{q^k}} \sin kt\) . The values of the best integral approximati...

We study a Nikol’skii type inequality for even entire functions of given exponential type between the uniform norm on the half-line [0,∞) and the norm (∫ 0∞|f(x)|q x2α+1dx)1/q of the space Lq ((0,∞), x2α+1) with the Bessel weight for 1 ≤ q < ∞ and α > −1/2. An extremal function is characterized. In particular, we prove that the uniform norm of an e...

July 16, 2018, was the 75th birthday of famous Russian scientist, prominent mathematician, Doctor of Physics and Mathematics, Professor Vitalii Vladimirovich Arestov.

We consider the problem of one-sided weighted integral approximation on the interval [−1, 1] to the characteristic functions of intervals (a, 1] ⊂ (−1, 1] and (a, b) ⊂ (−1, 1) by algebraic polynomials. In the case of half-intervals, the problem is solved completely. We construct an example to illustrate the difficulties arising in the case of an op...

We establish new estimates for the constant $J_a(k,\alpha)$ in the Brudnyi-Jackson inequality for approximation of $f \in C[-1,1]$ by algebraic polynomials: $$ E_{n}^a (f) \le J_a(k, \alpha) \ \omega_k (f, \alpha \pi /n ), \quad \alpha >0 $$ The main result of the paper implies the following inequalities $$ 1/2< J_a (2k, \alpha) < 10, \quad n \ge 2...

We consider the properties of functions f from the space L²(T) on the period T = [−π, π) with lacunary Fourier series such that the size of each gap is not less than a given positive integer q − 1. We find two-sided estimates of the L² norms of such functions on T in terms of similar norms (more exactly, seminorms) on intervals I of length |I| = 2h...

Let \(T_n^+\) be the set of nonnegative trigonometric polynomials \(\tau_n\) of degree \(n\) that are strictly positive at zero. For \(0\le\alpha\le2\pi/(n+2),\) we find the minimum of the mean value of polynomial \((\cos\alpha-\cos{x})\tau_n(x)/\tau_n(0)\) over \(\tau_n\in T_n^+\) on the period \([-\pi,\pi).\)

Best integral approximations of B-splines by trigonometric polynomials are found. A sharp Jackson type inequality with a special modulus of continuity is obtained

We consider a special 2k-order modulus of continuity W 2k (f,h) of 2π-periodic continuous functions and prove an analog of the Bernstein–Nikolsky–Stechkin inequality for trigonometric polynomials in terms of W 2k . We simplify the main construction from the paper by Foucart et al. (Constr. Approx. 29(2), 157–179, 2009) and give new upper estimates...

The value of the best one-sided integral approximation of the characteristic function of the interval (-h,h) by trigonometric polynomials of given degree is found for any 0<h≤π.

The present paper is devoted to the well-known problem of determining the maximum number of elementsτ m (s) of a sphericals-code (−1<-s<1) in Euclidean space ℝ m of dimensionm>-2; to be exact, here we consider the Delsarte functionw m (s) related toτ m (s) via the inequalityτ m (s) ≤w m (s). In this paper, the solution of the equationw m (s)=...

In 1935, Ya.L. Geronimus found the best integral approximation on the period [−π,π) of the function sin(n + 1)t − 2q sin nt, q ∈ ℝ, by the subspace of trigonometric polynomials of degree at most n − 1. This result is an integral analog of the known theorem by E.I. Zolotarev (1868). At present, there are several methods of proving this fact. We prop...

We apply the results on the integral approximation of the characteristic function of an interval by the subspace $$ \mathcal{T}_{n - 1} $$ of trigonometric polynomials of order at most n − 1, which were obtained by the authors earlier, to investigate the Jackson inequality between the best uniform approximation of a continuous periodic function by...

We prove that the value E n−1(χ h )L of the best integral approximation of the characteristic function χ h of an interval (−h, h) on the period [−π,π) by trigonometric polynomials of degree at most n − 1 is expressed in terms of zeros of the Bernstein function cos {nt − arccos[(2q − (1 + q 2) cost)/(1 + q 2 − 2q cost)]}, t ∈ [0, π], q ∈ (−1,1). Her...

This paper is devoted to the inequalities for mean values of functions from $T_{2n-1}^\perp$. The simple proof of the classical Jackson inequality in the case of the second modulus of continuity may be considered as the consequence of our estimates. The problems of the sharp constants in classical Stechkin's inequality are also discussed.

This note is a continuation of our papers [1,2], devoted to $L$-approximation of characteristic function of $(-h, h)$ by trigonometric polynomials. In the paper [1] the sharp values of the best approximation for the special values of $h$ were found. In [2] we gave the complete solution of the problem for arbitrary values of $h$. In general case [2]...

This paper is devoted to the equivalence of two type direct theorems in Approximation Theory: a) for smooth functions (Favard's estimates). b) for arbitrary continuous function (Jackson--Stechkin estimates). Specifically, we will show that Jackson--Stechkin inequality with optimal respect to the order of smoothness constants follows from Favard's i...

Let Lα,β² be the Hubert space of real-valued functions on [0, π] with scalar product (F,G) = ∫0π F(x)G(x) (sin x/2)2α+1 (cosx/2)2β+1 dx, α>-1,β>-1, and norm ∥F∥ = (F, F)1/2. We prove in the case when α > β ≥1/2 the following exact Jackson-Stechkin inequality between the best approximation of F by cosinepolynomials of order n -1 and its generalized...

In this paper we prove the Jackson-Stechkin inequalityE n−1(f)<ωn (f, 2τ n ,λ),n≥1,m≥5,r≥1, f ∈L2(\((\mathbb{S}^{m - 1} )\)),f ≢ const, which is sharp for eachn=2, 3, ...; hereE n−1 (f) is the best approximation of a functionf by spherical polynomials of degree ≤n−1, ωn (f, τ) is theτth modulus of continuity off based on the translations\(s_t f(x)...

We consider the Jackson-Stechkin inequality between the best mean square approximation of an arbitrary 2π-periodic complex-valued function from L 2 by trigonometric polynomials of a given degree and its modulus of continuity generated by a finite difference operator with coefficients depending continuously on the step of the operator. We obtain a l...