Related papers: Convexity properties related to extremal functions

We derive some key extremal features for $k$th order Markov chains that can be used to understand how the process moves between an extreme state and the body of the process. The chains are studied given that there is an exceedance of a…

A precise characterization of the extremal points of sublevel sets of nonsmooth penalties provides both detailed information about minimizers, and optimality conditions in general classes of minimization problems involving them. Moreover,…

Let $w_{\lambda}(t)=(1-t^2)^{\lambda-1/2}$, $\lambda>-1/2$, be the Gegenbauer weight function, and $\Vert\cdot\Vert$ denote the associated $L_2$-norm, i.e., $$ \Vert f\Vert:=\Big(\int_{-1}^{1}w_{\lambda}(t)\vert f(t)\vert^2\,dt\Big)^{1/2}.…

The Polyak-{\L}ojasiewicz (P{\L}) condition is often invoked in nonconvex optimization because it allows fast convergence of algorithms beyond strong convexity. A function $f \colon \mathcal{M} \to \mathbb{R}$ on a Riemannian manifold…

Let $\M$ be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (\ref{logfanhan}) has an extremal function decaying exponentially near infinity. We also…

Curves in the complex plane that satisfy the S-property were first introduced by Stahl and they were further studied by Gonchar and Rakhmanov in the 1980s. Rakhmanov recently showed the existence of curves with the S-property in a harmonic…

Let S(n) be the set of all polynomials of degree n with all roots in the unit disk, and define d(P) to be the maximum of the distances from each of the roots of a polynomial P to that root's nearest critical point. In this notation,…

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and…

Critical Sobolev-type inequality for a class of weighted Sobolev spaces on the entire space is established. We also investigate the existence of extremal function for the associated variational problem. As an application, we prove the…

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 consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We…

We show that differentiable functions, defined on a convex body $K \subseteq \mathbb R^d$, whose derivatives do not exceed a suitable given sequence of positive real numbers share many properties with polynomials. The role of the degree of…

This paper addresses both necessary and relevant sufficient extremum conditions for a variational problem defined by a smooth Lagrangian, involving higher derivatives of several variable vector valued functions. A general formulation of…

The present paper is a continuation of the author's previous works, in which necessary and sufficient local extrema at a stationary point of a polynomial or a power series (and thus of an analytic function) are given. It is known that for…

Extremal functions for the $n$th coefficient in the Krzy\.z conjecture are atomic singular inner functions with at most $n$ atoms. This paper gives a lower bound on the number of atoms $N$ of the form $N\geq cn$, marking progress toward…

Let $E$ be a compact set of positive logarithmic capacity in the complex plane and let $\{P_n(z)\}_{1}^{\infty}$ be a sequence of asymptotically extremal monic polynomials for $E$ in the sense that \begin{equation*}%\label{}…

An important problem in applications of quasiconformal analysis and in its numerical aspect is to establish algorithms for explicit or approximate determination of the basic quasiinvariant curvelinear and analytic functionals intrinsically…

We generalized the Korkin-Zolotarev theorem to the case of entire functions having the smallest $L^1$ norm on a system of intervals $E$. If $\bbC\setminus E$ is a domain of Widom type with the Direct Cauchy Theorem we give an explicit…

We show a large class of analytic submanifolds of C^n to be strongly extremal. This generalizes V. Sprindzhuk's solution of the complex case of Mahler's Problem, and settles complex analogues of conjectures made in the 1970s by Baker and…

We introduce an asymptotic Markov's exponent and show that it is equal to Markov's exponent for a wide class of norms. As a consequence we obtain a lower bound for the optimal exponent in Markov's inequality considered with the norms…