Expand all Collapse all | Results 1 - 4 of 4 |
1. CJM 2011 (vol 63 pp. 1025)
Universal Series on a Riemann Surface Every holomorphic function on a compact subset of a Riemann surface can
be uniformly approximated by partial sums of a given series of functions.
Those functions behave locally like the classical fundamental solutions
of the Cauchy-Riemann operator in the plane.
Categories:30B60, 30E10, 30F99 |
2. CJM 2002 (vol 54 pp. 945)
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications |
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications Given a homogeneous elliptic partial differential operator $L$ with constant
complex coefficients and a class of functions (jet-distributions) which
are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and
which belong locally to a Banach space $V$, we consider the problem of
approximating in the norm of $V$ the functions in this class by ``analytic''
and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth,
Arakelyan (including tangential) and Carleman type theorems for a large class
of Banach spaces $V$ and operators $L$. Important applications to boundary
value problems of solutions of homogeneous elliptic partial differential
equations are obtained, including the solution of a generalized Dirichlet
problem.
Keywords:approximation on closed sets, elliptic operator, strongly elliptic operator, $L$-meromorphic and $L$-analytic functions, localization operator, Banach space of distributions, Dirichlet problem Categories:30D40, 30E10, 31B35, 35Jxx, 35J67, 41A30 |
3. 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 |
4. CJM 1999 (vol 51 pp. 117)
Meromorphic functions with prescribed asymptotic behaviour, zeros and poles and applications in complex approximation |
Meromorphic functions with prescribed asymptotic behaviour, zeros and poles and applications in complex approximation We construct meromorphic functions with asymptotic power series
expansion in $z^{-1}$ at $\infty$ on an Arakelyan set $A$ having
prescribed zeros and poles outside $A$. We use our results to prove
approximation theorems where the approximating function fulfills
interpolation restrictions outside the set of approximation.
Keywords:asymptotic expansions, approximation theory Categories:30D30, 30E10, 30E15 |