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Search: MSC category 31A15 ( Potentials and capacity, harmonic measure, extremal length [See also 30C85] )

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1. CJM 2005 (vol 57 pp. 1080)

Pritsker, Igor E.
The Gelfond--Schnirelman Method in Prime Number Theory
The original Gelfond--Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on $[0,1]$ to give a Chebyshev-type lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's $\psi$-function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomial-type weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support.

Keywords:distribution of prime numbers, polynomials, integer, coefficients, weighted transfinite diameter, weighted capacity, potentials
Categories:11N05, 31A15, 11C08

2. CJM 2003 (vol 55 pp. 576)

Lukashov, A. L.; Peherstorfer, F.
Automorphic Orthogonal and Extremal Polynomials
It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szeg\"o polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szeg\"o polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients {\it etc.} are derived.

Categories:42C05, 30F35, 31A15, 41A21, 41A50

3. CJM 2002 (vol 54 pp. 225)

Arslan, Bora; Goncharov, Alexander P.; Kocatepe, Mefharet
Spaces of Whitney Functions on Cantor-Type Sets
We introduce the concept of logarithmic dimension of a compact set. In terms of this magnitude, the extension property and the diametral dimension of spaces $\calE(K)$ can be described for Cantor-type compact sets.

Categories:46E10, 31A15, 46A04

4. CJM 2000 (vol 52 pp. 815)

Lubinsky, D. S.
On the Maximum and Minimum Modulus of Rational Functions
We show that if $m$, $n\geq 0$, $\lambda >1$, and $R$ is a rational function with numerator, denominator of degree $\leq m$, $n$, respectively, then there exists a set $\mathcal{S}\subset [0,1] $ of linear measure $\geq \frac{1}{4}\exp (-\frac{13}{\log \lambda })$ such that for $r\in \mathcal{S}$, \[ \max_{|z| =r}| R(z)| / \min_{|z| =r} | R(z) |\leq \lambda ^{m+n}. \] Here, one may not replace $\frac{1}{4}\exp ( -\frac{13}{\log \lambda })$ by $\exp (-\frac{2-\varepsilon }{\log \lambda })$, for any $\varepsilon >0$. As our motivating application, we prove a convergence result for diagonal Pad\'{e} approximants for functions meromorphic in the unit ball.

Categories:30E10, 30C15, 31A15, 41A21

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