Polynomial almost periodic solutions for a class of Riemann–Hilbert problems with triangular symbols
Let g ˆ ( ξ ) = a e i α ξ + b + c e − i β ξ with α , β ∈ ] 0 , 1 [ such that α + β < 1 , α β −1 ∉ Q and a , b , c ∈ C ∖ { 0 } . In this paper the existence of almost-periodic polynomial (APP) solutions to the equation g ˆ h + = E l + + l − (with h + ∈ H ∞ + ∩ E H ∞ − and l ± ∈ H ∞ ± ) is studied....
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| Published in: | Journal of functional analysis Vol. 240; no. 1; pp. 226 - 268 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier Inc
01-11-2006
|
| Subjects: | |
| Online Access: | Get full text |
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| Summary: | Let
g
ˆ
(
ξ
)
=
a
e
i
α
ξ
+
b
+
c
e
−
i
β
ξ
with
α
,
β
∈
]
0
,
1
[
such that
α
+
β
<
1
,
α
β
−1
∉
Q
and
a
,
b
,
c
∈
C
∖
{
0
}
. In this paper the existence of almost-periodic polynomial (APP) solutions to the equation
g
ˆ
h
+
=
E
l
+
+
l
−
(with
h
+
∈
H
∞
+
∩
E
H
∞
−
and
l
±
∈
H
∞
±
) is studied. The natural space in which to seek a solution to the above problem is the space of almost periodic functions with spectrum in the group
α
Z
+
β
Z
+
Z
. Due to the difficulty in dealing with the problem in that generality, solutions are sought with spectrum in the group
α
Z
+
β
Z
. Several interesting and totally new results are obtained. It is shown that, if
1
∉
α
Z
+
β
Z
, no polynomial solutions exist, i.e., almost periodic polynomial solutions exist only if
α
Z
+
β
Z
=
α
Z
+
β
Z
+
Z
. Keeping to this setting, it is shown that APP solutions exist if and only if the function
g
ˆ
satisfies the simple spectral condition
α
+
β
>
1
/
2
. The proof of this result is nontrivial and has a number-theoretic flavour. Explicit formulas for the solution to the above problem are given in the final section of the paper. The derivation of these formulas is to some extent a byproduct of the proof of the result on the existence of APP solutions. |
|---|---|
| ISSN: | 0022-1236 1096-0783 |
| DOI: | 10.1016/j.jfa.2005.12.022 |