Analytic approximation of matrix functions in L p
We consider the problem of approximation of matrix functions of class L p on the unit circle by matrix functions analytic in the unit disk in the norm of L p , 2 ≤ p < ∞ . For an m × n matrix function Φ in L p , we consider the Hankel operator H Φ : H q ( C n ) → H − 2 ( C m ) , 1 / p + 1 / q = 1...
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Published in | Journal of approximation theory Vol. 158; no. 2; pp. 242 - 278 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Elsevier Inc
01.06.2009
|
Online Access | Get full text |
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Summary: | We consider the problem of approximation of matrix functions of class
L
p
on the unit circle by matrix functions analytic in the unit disk in the norm of
L
p
,
2
≤
p
<
∞
. For an
m
×
n
matrix function
Φ
in
L
p
, we consider the Hankel operator
H
Φ
:
H
q
(
C
n
)
→
H
−
2
(
C
m
)
,
1
/
p
+
1
/
q
=
1
/
2
. It turns out that the space of
m
×
n
matrix functions in
L
p
splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If
Φ
is respectable, then its distance to the set of analytic matrix functions is equal to the norm of
H
Φ
. For weird matrix functions, to obtain the distance formula, we consider Hankel operators defined on spaces of matrix functions. We also describe the set of
p
-badly approximable matrix functions in terms of special factorizations and give a parametrization formula for all best analytic approximants in the norm of
L
p
. Finally, we introduce the notion of
p
-superoptimal approximation and prove the uniqueness of a
p
-superoptimal approximant for rational matrix functions. |
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ISSN: | 0021-9045 1096-0430 |
DOI: | 10.1016/j.jat.2008.08.009 |