Expand all Collapse all | Results 1 - 4 of 4 |
1. CJM 2005 (vol 57 pp. 1080)
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)
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)
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)
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 |