Related papers: On C. Michel's hypothesis about the modulus of typ…

In the class of normalized sine-polynomials $S(t),$ non-negative on $[0,\pi],$ W.Rogosinski and G.Szeg\H{o} 1950 considered a number of extremal problems and proved, among other things, sharp upper and lower estimates for the coefficient…

We present a survey of central developments in the theory of Chebyshev polynomials, introduced by P.~L.~Chebyshev and later extended to the complex plane by G.~Faber. Our primary focus is their defining extremal property: among all…

For the class of sine polynomials $b_1\sin t+b_2\sin2t+...+b_N\sin Nt,\; (b_N\not= 0),$ which are nonnegative on $(0,\pi)$, W. Rogosinski and G. Szeg\"o derived, among other things, exact bounds for $|b_2|$ via the Luk\'acs presentation of…

In approximation theory, logarithmic derivatives of complex polynomials are called simple partial fractions (SPF) as suggested by Eu.P. Dolzhenko. Many solved and unsolved extremal problems related to SPF are traced back to works of G.…

For the univalent polynomials $F(z) = \sum\limits_{j=1}^{N} a_j z^{2j-1}$ with real coefficients and normalization \(a_1 = 1\) we solve the extremal problem \[ \min_{a_j:\,a_1=1} \left( -iF(i) \right) = \min_{a_j:\,a_1=1}…

In 1962 J. Siciak introduced polynomial extremal function. The Siciak extremal function is one of the most important tool to investigate derivatives of polynomials. W. Ple\'sniak in the circle of papers (most important joint with W.…

For a pair of conjugate trigonometrical polynomials $C (t) = \sum_ { j = 1 } ^N { { a_j}\cos jt }, S(t) = \sum_ { j = 1 } ^N { { a_j}\sin jt }$ with real coefficients and normalization ${a_1} = 1 $ we solve the extremal problem \[ \sup_…

The standard well-known Remez inequality gives an upper estimate of the values of polynomials on $[-1,1]$ if they are bounded by $1$ on a subset of $[-1,1]$ of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev…

In the paper we generalize the notion of problem (P) introduced by Poletsky. We introduce the notion of (P_m) extremals. For example, geodesics are (P_1) extremals. Using obtained results we present a description of (P_m) extremals in…

We consider polynomials of degree $d$ with only real roots and a fixed value of discriminant, and study the problem of minimizing the absolute value of polynomials at a fixed point off the real line. There are two explicit families of…

We consider the classical problem of estimating norms of higher order derivatives of algebraic polynomial via the norms of polynomial itself. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A.…

We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials.…

We consider the classical problem of estimating norm of the derivative of algebraic polynomial via the norm of polynomial itself. The corresponding extremal problem for general polynomials in uniform norm was solved by V. Markov. In this…

This paper is devoted to Markov's extremal problems of the form $M_{n,k}=\sup_{p\in\PP_n\setminus\{0\}}{{\|p^{(k)}\|}_X}/{{\|p\|}_X}$ $(1\le k\le n)$, where $\PP_n$ is the set of all algebraic polynomials of degree at most $n$ and $X$ is a…

The century old extremal problem, solved by Carath\'eodory and Fej\'er, concerns a nonnegative trigonometric polynomial normalized by a0 = 1, and the quantity to be maximized is the coefficient a1. In the complex exponential form, the…

We consider the problem of maximizing the sum of squares of the leading coefficients of polynomials $P_{i_1}(x),\ldots ,P_{i_m}(x)$ (where $P_j(x)$ is a polynomial of degree $j$) under the restriction that the sup-norm of $\sum_{j=1}^m…

In this article the solution of the special problem of the conditional extremum for the conjugate trigonometric polynomials is given. A possibility to apply this result to the problems of optimal stabilization of quasidynamic chaos in…

In this paper, we consider the Atkin-like polynomials that appeared in the study of normalized extremal quasimodular forms of depth 1 on $SL_{2}(\mathbb{Z})$ by Kaneko and Koike as orthogonal polynomials and clarify their properties. By…

We show that the exceptional orthogonal polynomials can be viewed as confluent limits of the generalized Schur polynomials introduced by Sergeev and Veselov.

We investigate the weighted Bojanov-Chebyshev extremal problem for trigonometric polynomials, that is, the minimax problem of minimizing $\|T\|_{w,C({\mathbb T})}$, where $w$ is a sufficiently nonvanishing, upper bounded, nonnegative weight…