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1. CJM 2009 (vol 61 pp. 1341)
Simultaneous Polynomial Approximations of the Lerch Function We construct bivariate polynomial approximations of the Lerch
function that for certain specialisations of the variables and
parameters turn out to be Hermite--Pad\'e approximants either of
the polylogarithms or of Hurwitz zeta functions. In the former
case, we recover known results, while in the latter the results
are new and generalise some recent works of Beukers and Pr\'evost.
Finally, we make a detailed comparison of our work with Beukers'.
Such constructions are useful in the arithmetical study of the
values of the Riemann zeta function at integer points and of the
Kubota--Leopold $p$-adic zeta function.
Categories:41A10, 41A21, 11J72 |
2. CJM 2006 (vol 58 pp. 249)
Convergence of Fourier--PadÃ© Approximants for Stieltjes Functions We prove convergence of diagonal multipoint Pad\'e approximants of
Stieltjes-type functions when a certain moment problem is
determinate. This is used for the study of the convergence of
Fourier--Pad\'e and nonlinear Fourier--Pad\'e approximants for such
type of functions.
Keywords:rational approximation, multipoint PadÃ© approximants, Fourier--PadÃ© approximants, moment problem Categories:41A20, 41A21, 44A60 |
3. 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 |
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 |