Geno Nikolov

Ana Avdzhieva, Geno Nikolov

We construct several sequences of asymptotically optimal definite quadrature formulae of fourth order and evaluate their error constants. Besides the asymptotical optimality, an advantage of our quadrature formulae is the explicit form of their weights and nodes. For the remainders of our quadrature formulae monotonicity properties are established when the integrand is a 4-convex function, and a-posteriori error estimates are proven.

Geno Nikolov, Alexei Shadrin

We are continuing out studies of the so-called Markov inequalities with a majorant. Inequalities of this type provide a bound for the $k$-th derivative of an algebraic polynomial when the latter is bounded by a certain curved majorant $μ$. A conjecture is that the upper bound is attained by the so-called snake-polynomial which oscillates most between $\pm μ$, but it turned out to be a rather difficult question. In the previous paper, we proved that this is true in the case of symmetric majorant provided the snake-polynomial has a positive Chebyshev expansion. In this paper, we show that that the conjecture is valid under the condition of positive expansion only, hence for non-symmetric majorants as well.