2019
Том 71
№ 11

# Bondarenko A. V.

Articles: 3
Brief Communications (English)

### On a spherical code in the space of spherical harmonics

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 857 – 859

We propose a new method for the construction of new “nice” configurations of vectors on the unit sphere $S^d$ with the use of spaces of spherical harmonics.

Brief Communications (Russian)

### Negative result in pointwise 3-convex polynomial approximation

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 563-567

Let $Δ^3$ be the set of functions three times continuously differentiable on $[−1, 1]$ and such that $f'''(x) ≥ 0,\; x ∈ [−1, 1]$. We prove that, for any $n ∈ ℕ$ and $r ≥ 5$, there exists a function $f ∈ C^r [−1, 1] ⋂ Δ^3 [−1, 1]$ such that $∥f (r)∥_{C[−1, 1]} ≤ 1$ and, for an arbitrary algebraic polynomial $P ∈ Δ^3 [−1, 1]$, there exists $x$ such that $$|f(x)−P(x)| ≥ C \sqrt{n}ρ^r_n(x),$$ where $C > 0$ is a constant that depends only on $r, ρ_n(x) := \frac1{n^2} + \frac1n \sqrt{1−x^2}$.

Article (Ukrainian)

### Investigation of one class of diophantine equations

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 831–836

We consider the problem of existence of solutions of the equation $\frac{X}{Y} + \frac{Y}{Z} + \frac{Z}{X} = m$ in natural numbers for differentmN. We prove that this equation possesses solutions in natural numbers form=a 2+5,aZ, and does not have solutions ifm=4p 2,pN, andp is not divisible by 3. We also prove that, forn≥12, the equation $$\frac{{b_1 }}{{b_2 }} + \frac{{b_2 }}{{b_3 }} + \cdots + \frac{{b_{n - 1} }}{{b_n }} + \frac{{b_n }}{{b_1 }} = m$$ possesses solutions in natural numbers if and only ifmn,mN.