Point-wise estimates for the derivative of algebraic polynomials

We give a sufficient condition on coefficients $a_k$ of an algebraic polynomial $P(z)=\sum\limits_{k=0}^{n}a_kz^k$, $a_n\not=0,$ such that the pointwise Bernstein inequality $|P'(z)|\le n|P(z)|$ is true for all $z,\ |z|\le 1$.

Inequalities for the First and Second Derivatives of Algebraic Polynomials on an Ellipse

We prove a theorem on the preservation of inequalities between functions of a special form after differentiation on an ellipse. In particular, we obtain generalizations of the Duffin–Schaeffer inequality and the Vidensky inequality for the first and second derivatives of algebraic polynomials to an ellipse.

On approximation of nonperiodic functions by algebraic polynomials in $L_p$ metric ($0 < p < 1$)

In the paper, we consider approximations of nonperiodic functions defined on $[-1, 1]$ by algebraic polynomials in $L_p$ metric ($0 < p < 1$).In particular, for some classes we provide the constructive characteristic in the same metric.

Exact values of the best mean approximations by algebraic polynomials of $W^r_L$ classes ($r=1,2$)

In the paper, we have found the supremum of the best mean approximations by algebraic polynomials of differentiable functions from $W^r_L$ classes for $r=1,2$.

One-sided approximation of some classes of singular integrals with regard to point position on the interval

We obtain asymptotically exact estimates of approximation of functions from some classes of singular integrals by algebraic polynomials with regard to point position on the interval.

On approximation of functions by algebraic polynomials in $L^p_{\rho}$ metric

The version has been found, of improved Jackson's theorem analogue, concerning the approximation by algebraic polynomials on the interval in the integral metric for some classes of functions being integrable with the following weight: $\rho(x) = (1-x)^{\alpha} (1+x)^{\beta}$.

The best one-side approximation of classes of integrals of fractional order with regard to position of point on interval

We obtain the asymptotic estimations for the best one-sided point-wise approximation to the classes $W_{\infty}^r$, $r > 1$ (in case of fractional $r$) by algebraic polynomials.