# Dzyubenko H. A.

### Almost coconvex approximation of continuous periodic functions

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 353-367

If a $2\pi$ -periodic function $f$ continuous on the real axis changes its convexity at $2s, s \in N$, points $y_i : \pi \leq y_{2s} < y_{2s-1} < . . . < y_1 < \pi$ , and, for all other $i \in Z$, $y_i$ are periodically defined, then, for every natural $n \geq N_{y_i}}$, we determine a trigonometric polynomial $P_n$ of order cn such that $P_n$ has the same convexity as $f$ everywhere except, possibly, small neighborhoods of the points $y_i : (y_i \p_i /n, y_i + \pi /n)$, and $\| f P_n\| \leq c(s) \omega 4(f, \pi /n)$,, where $N_{y_i}}$ is a constant depending only on $\mathrm{m}\mathrm{i}\mathrm{n}_{i = 1,...,2s}\{ y_i y_{i+1}\} , c$ and $c(s)$ are constants depending only on $s, \omega 4(f, \cdot )$ is the fourth modulus of smoothness of the function $f$, and $\| \cdot \|$ is the max-norm.

### Pointwise estimation of an almost copositive approximation of continuous functions by algebraic polynomials

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 641-649

In the case where a function continuous on a segment $f$ changes its sign at $s$ points $y_i : 1 < y_s < y_{s-1} < ... < y_1 < 1$, for any $n \in N$ greater then a constant $N(k, y_i)$ that depends only on $k \in N$ and \$\min_{i=1,...,s-1}\{ y_i - y_{i+1}\}$, we determine an algebraic polynomial $P_n$ of degree \leq n such that: $P_n$ has the same sign as f everywhere except possibly small neighborhoods of the points $y_i$: ($$(y_i \rho_n(y_i), y_i + \rho_n(y_i)),\quad \rho_n(x) := 1/n2 + \sqrt{1 - x^2}/n,$$ $P_n(y_i) = 0$ and $$| f(x) P_n(x)| \leq c(k, s)\omega_k(f, \rho_n(x)),\quad x \in [ 1, 1],$$ where $c(k, s)$ is a constant that depends only on $k$ and $s$ and $\omega k(f, \cdot )$ is the modulus of continuity of the function $f$ of order $k$.

### Comonotone approximation of twice differentiable periodic functions

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 435-451

In the case where a $2π$-periodic function $f$ is twice continuously differentiable on the real axis $ℝ$ and changes its monotonicity at different fixed points $y_i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ $(i.e., on $ℝ$, there exists a set $Y := {y_i } i∈ℤ$ of points $y_i = y_{i+2s} + 2π$ such that the function $f$ does not decrease on $[y_i , y_{i−1}]$ if $i$ is odd and does not increase if $i$ is even), for any natural $k$ and $n, n ≥ N(Y, k) = const$, we construct a trigonometric polynomial $T_n$ of order $≤n$ that changes its monotonicity at the same points $y_i ∈ Y$ as $f$ and is such that $$∥f−T_n∥ ≤ \frac{c(k,s)}{n^2} ω_k(f″,1/n)$$ $$(∥f−T_n∥ ≤ \frac{c(r+k,s)}{n^r} ω_k(f^{(r)},1/ n),f ∈ C^{(r)},\; r ≥ 2),$$ where $N(Y, k)$ depends only on $Y$ and $k, c(k, s)$ is a constant depending only on $k$ and $s, ω k (f, ⋅)$ is the modulus of smoothness of order $k$ for the function $f$, and $‖⋅‖$ is the max-norm.

### Pointwise Estimates for the Coconvex Approximation of Differentiable Functions

Dzyubenko H. A., Zalizko V. D.

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 47–59

We obtain pointwise estimates for the coconvex approximation of functions of the class $W^r,\; r > 3$.

### Coconvex Approximation of Functions with More than One Inflection Point

Dzyubenko H. A., Zalizko V. D.

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 352-365

Assume that *f* ∈ *C*[−1, 1] belongs to *C*[−1, 1] and changes its convexity at *s* > 1 different points *y* _{i}, \(\overline {1,s} \) , from (−1, 1). For *n* ∈ *N*, *n* ≥ 2, we construct an algebraic polynomial *P* _{n} of order ≤ *n* that changes its convexity at the same points *y* _{i} as *f* and is such that $$|f(x) - P_n (x)|\;\; \leqslant \;\;C(Y)\omega _3 \left( {f;\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right),\;\;\;\;\;x\;\; \in \;\;[ - 1,\;1],$$ where ω_{3}(*f*; *t*) is the third modulus of continuity of the function *f* and *C*(*Y*) is a constant that depends only on \(\mathop {\min }\limits_{i = 0,...,s} \left| {y_i - y_{i + 1} } \right|,\;\;y_0 = 1,\;\;y_{s + 1} = - 1\) , *y* _{0} = 1, *y* _{ s + 1} = −1.

### Coconvex Pointwise Approximation

Dzyubenko H. A., Gilewicz J., Shevchuk I. A.

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1200-1212

Assume that a function *f* ∈ *C*[−1, 1] changes its convexity at a finite collection *Y* := {*y* _{1}, ... *y* _{s}} of *s* points *y* _{i} ∈ (−1, 1). For each *n* > *N*(*Y*), we construct an algebraic polynomial *P* _{n} of degree ≤ *n* that is coconvex with *f*, i.e., it changes its convexity at the same points *y* _{i} as *f* and $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c{\omega }_{2} \left( {f,\frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in \left[ { - 1,1} \right],$$ where *c* is an absolute constant, ω_{2}(*f*, *t*) is the second modulus of smoothness of *f*, and if *s* = 1, then *N*(*Y*) = 1. We also give some counterexamples showing that this estimate cannot be extended to the case of higher smoothness.

### Copositive pointwise approximation

Ukr. Mat. Zh. - 1996. - 48, № 3. - pp. 326-334

We prove that if a function*f* ∈*C* ^{(1)} (*I*),*I*: = [−1, 1], changes its sign*s* times (*s* ∈ ℕ) within the interval*I*, then, for every*n* > *C*, where*C* is a constant which depends only on the set of points at which the function changes its sign, and*k* ∈ ℕ, there exists an algebraic polynomial*P* _{ n } =*P* _{ n }(*x*) of degree ≤*n* which locally inherits the sign of*f(x)* and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω_{ k } (*f*′;*t*) is the*k*th modulus of continuity of the function*f*’. It is also shown that if*f* ∈*C* (*I*) and*f*(*x*) ≥ 0,*x* ∈*I* then, for any*n* ≥*k* − 1, there exists a polynomial*P* _{ n } =*P* _{ n } (*x*) of degree ≤*n* such that*P* _{ n } (*x*) ≥ 0,*x* ∈*I*, and |*f*(*x*) −*P* _{ n }(*x*)| ≤*c*(*k*)ω_{ k } (*f*;*n* ^{−2} +*n* ^{−1} √1 −*x* ^{2}),*x* ∈*I*.

### Pointwise estimation of comonotone approximation

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1467–1472

We prove that, for a continuous function *f(x)* defined on the interval [−1,1] and having finitely many intervals where it is either nonincreasing or nondecreasing, one can always find a sequence of polynomials *P* _{ n } *(x)* with the same local properties of monotonicity as the function *f(x)* and such that ¦*f(x)*−*P* _{ n } *(x)* ¦≤*C*ω_{2}(*f*;n^{−2}+*n* ^{−1}√1−*x* ^{2}), where*C* is a constant that depends on the length of the smallest interval.

### Uniform estimates for monotonic polynomial approximation

Dzyubenko H. A., Listopad V. V., Shevchuk I. A.

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 38–43

The uniform estimate is established for a monotone polynomial approximation of functions whose smoothness decreases at the ends of a segment.