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An Extremal Problem for the Bergman Kernel of Orthogonal Polynomials
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Journal Article
An Extremal Problem for the Bergman Kernel of Orthogonal Polynomials
2025
Overview
Let
Γ
⊂
C
be a curve of class
C
(
1
,
α
)
. For
z
0
in the unbounded component of
C
\\
Γ
, and for
n
=
1
,
2
,
.
.
.
, let
ν
n
be a probability measure with
supp
(
ν
n
)
⊂
Γ
which minimizes the Bergman function
B
n
(
ν
,
z
)
:
=
∑
k
=
0
n
|
q
k
ν
(
z
)
|
2
at
z
0
among all probability measures
ν
on
Γ
(here,
{
q
0
ν
,
…
,
q
n
ν
}
are an orthonormal basis in
L
2
(
ν
)
for the holomorphic polynomials of degree at most
n
). We show that
{
ν
n
}
n
tends weak-* to
δ
^
z
0
, the balayage of the point mass at
z
0
onto
Γ
, by relating this to an optimization problem for probability measures on the unit circle. Our proof makes use of estimates for Faber polynomials associated to
Γ
.
Publisher
Springer US,Springer Nature B.V,Springer Verlag
Subject
