Weighted energy problem on the unit circle
We solve the weighted energy problem on the unit circle, by finding the extremal measure and describing its support. Applications to polynomial and exponential weights are also included.
math.CV↗
arXiv subjects
Igor E. Pritsker
Publications and source records attributed to Igor E. Pritsker.
How to find a measure from its potential
We consider the problem of finding a measure from the given values of its logarithmic potential on the support. It is well known that a solution to this problem is given by the generalized Laplacian. The case of our main interest is when the support is contained in a rectifiable curve, and the measure is absolutely continuous with respect to the arclength on this curve. Then the generalized Laplacian is expressed by a sum of normal derivatives of the potential. Such representation was available for smooth curves, and we show it holds for any rectifiable curve in the plane. We also relax the assumptions imposed on the potential. Finding a measure from its potential often leads to another closely related problem of solving a singular integral equation with Cauchy kernel. The theory of such equations is well developed for smooth curves. We generalize this theory to the class of Ahlfors regular curves and arcs, and characterize the bounded solutions on arcs.
math.CV↗
Convergence of Julia polynomials
We study the approximation of conformal mappings with the polynomials defined by Keldysh and Lavrentiev from an extremal problem considered by Julia. These polynomials converge uniformly on the closure of any Smirnov domain to the conformal mapping of this domain onto a disk. We prove estimates for the rate of such convergence on domains with piecewise analytic boundaries, expressed through the smallest exterior angles at the boundary.
math.CV↗
Small polynomials with integer coefficients
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. Their factors, zero distribution and asymptotics are the main subjects of this paper. In particular, we show that the integer Chebyshev polynomials for any infinite subset of the real line must have infinitely many distinct factors, which answers a question of Borwein and Erdélyi. Furthermore, it is proved that the accumulation set for their zeros must be of positive capacity in this case. We also find the first nontrivial examples of explicit integer Chebyshev constants for certain classes of lemniscates. Since it is rarely possible to obtain an exact value of integer Chebyshev constant, good estimates are of special importance. Introducing the methods of weighted potential theory, we generalize and improve the Hilbert-Fekete upper bound for integer Chebyshev constant. These methods also give bounds for the multiplicities of factors of integer Chebyshev polynomials, and lower bounds for integer Chebyshev constant. Moreover, all the mentioned bounds can be found numerically, by using various extremal point techniques, such as weighted Leja points algorithm. Applying our results in the classical case of the segment $[0,1]$, we improve the known bounds for the integer Chebyshev constant and the multiplicities of factors of the integer Chebyshev polynomials.
math.CA↗
Norms of products and factors polynomials
We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. Generalizing Gel'fond-Mahler inequality for the unit disk and Kneser-Borwein inequality for the segment $[-1,1]$, we prove an asymptotically sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set in plane. Applying similar techniques, we produce a related inequality for the norm of a single monic factor of a monic polynomial. The best constants in both inequalities are obtained by potential theoretic methods. We also consider applications of the general results to the cases of a disk and a segment.
math.CV↗
An inequality for the norm of a polynomial factor
Let $p(z)$ be a monic polynomial of degree $n$, with complex coefficients, and let $q(z)$ be its monic factor. We prove an asymptotically sharp inequality of the form $\|q\|_{E} \le C^n \|p\|_E$, where $\|\cdot\|_E$ denotes the sup norm on a compact set $E$ in the plane. The best constant $C_E$ in this inequality is found by potential theoretic methods. We also consider applications of the general result to the cases of a disk and a segment.
math.CV↗