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Department of Mathematics and Statistics
of University of Helsinki

Consider
the weighted Lpspace on the open
unit disc D of the complex plane,
where the weight decreases rapidly, say, exponentially, with the
boundary distance. In case p = 2 there still exists the
orthogonal projection Q onto the subspace of analytic functions.
However, it is known that Q is not bounded for the
weighted Lpnorms, if p≠2.
No other usual Bergmantype projection is known to be
bounded in these case either.
SOME RECENT ARTICLES

W.Lusky, J.Taskinen:
Bounded holomorphic projections for exponentially
decreasing weights.
J. Function Spaces Appl. 6, 1 (2008), 5970.
We construct bounded projection operators (in the sense of
Banach space operator theory, i.e., operators satisfying the idempotence
P = PP) from the weighted
L∞space onto its subspace consisting on analytic functions.
The domain is the open unit disc of the complex plain, and the weights
include cases decreasing exponentially with the boundary distance.
(The weights form a class called (B).)
The construction is based on Lusky's paper in Studia Math. 175 (2006).
It is more easily explained, if one only takes a projection P
from the weighted space of continuous functions f onto the analytic
ones:
Each continuous function on the disc has a Fourierseries on each
fixed radius r = z < 1 . The projection is constructed
using Cesarosummation methods to these Fourier series in a way which depends
on r: then Pf will be a Taylor series
such that the higher order terms come from the Fourierseries corresponding
to r close to 1.
We do not know an integral formula for the operator. Those presented
in the
Studia Math. (2005)paper with Bonet and Engliš
do not coincide with this operator
(see Formula 2 on the corresponding
page).

W.Lusky, J.Taskinen:
On weighted spaces of holomorphic functions of
several variables.
To appear in Israel J.Math.
We give a nontrivial generalization of the previous paper to
the domain which is the unit ball in higher complex dimensions.
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