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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 2000 (vol 52 pp. 982)
Holomorphic Functions of Slow Growth on Nested Covering Spaces of Compact Manifolds Let $Y$ be an infinite covering space of a projective manifold
$M$ in $\P^N$ of dimension $n\geq 2$. Let $C$ be the intersection with
$M$ of at most $n-1$ generic hypersurfaces of degree $d$ in $\mathbb{P}^N$.
The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $\phi$
be the smoothed distance from a fixed point in $Y$ in a metric pulled up
from $M$. Let $\O_\phi(X)$ be the Hilbert space of holomorphic
functions $f$ on $X$ such that $f^2 e^{-\phi}$ is integrable on $X$, and
define $\O_\phi(Y)$ similarly. Our main result is that (under more
general hypotheses than described here) the restriction $\O_\phi(Y)
\to \O_\phi(X)$ is an isomorphism for $d$ large enough.
This yields new examples of Riemann surfaces and domains of holomorphy
in $\C^n$ with corona. We consider the important special case when $Y$
is the unit ball $\B$ in $\C^n$, and show that for $d$ large enough,
every bounded holomorphic function on $X$ extends to a unique function
in the intersection of all the nontrivial weighted Bergman spaces on
$\B$. Finally, assuming that the covering group is arithmetic, we
establish three dichotomies concerning the extension of bounded
holomorphic and harmonic functions from $X$ to $\B$.
Categories:32A10, 14E20, 30F99, 32M15 |
3. CJM 1998 (vol 50 pp. 547)
Mittag-Leffler theorems on Riemann surfaces and Riemannian manifolds Cauchy and Poisson integrals over {\it unbounded\/} sets are employed to
prove Mittag-Leffler type theorems with massive singularities as well as
approximation theorems for holomorphic and harmonic functions.
Keywords:holomorphic, harmonic, Mittag-Leffler, Runge Categories:30F99, 31C12 |