On Banach frameness of degenerate weighted exponential system
This work deals with the frameness of weighted exponential system E ( ω , Z ) = { ω ( t ) e i n t } n ∈ Z in the space L p ( − π , π ) , p > 1 , with the weight function ω ( t ) of general form. Basis properties of E ( ω , Z ) in L p ( − π , π ) , p > 1 , are studied, in other words, the crite...
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| Published in | Fixed point theory and algorithms for sciences and engineering Vol. 2025; no. 1; pp. 26 - 15 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
23.09.2025
Springer Nature B.V SpringerOpen |
| Subjects | |
| Online Access | Get full text |
| ISSN | 2730-5422 2730-5422 |
| DOI | 10.1186/s13663-025-00805-5 |
Cover
| Abstract | This work deals with the frameness of weighted exponential system
E
(
ω
,
Z
)
=
{
ω
(
t
)
e
i
n
t
}
n
∈
Z
in the space
L
p
(
−
π
,
π
)
,
p
>
1
, with the weight function
ω
(
t
)
of general form. Basis properties of
E
(
ω
,
Z
)
in
L
p
(
−
π
,
π
)
,
p
>
1
, are studied, in other words, the criteria of completeness, minimality and basicity of the system
E
(
ω
,
Z
)
in the space
L
p
(
−
π
,
π
)
,
p
>
1
, are given. Sufficient conditions for the completeness and minimality of
E
(
ω
,
Z
∖
F
)
in
L
p
(
−
π
,
π
)
,
p
>
1
, are found, where
F
is an arbitrary finite nonempty subset of the set of integers
Z
. A different method to prove that the system
E
(
ω
,
Z
∖
F
)
does not form a Schauder basis for
L
p
(
−
π
,
π
)
,
p
>
1
, is given. Theorem on a property of expansion system and criterion of Banach frameness for
E
(
ω
,
Z
)
in
L
p
(
−
π
,
π
)
,
p
>
1
, are proved. In particular, it is proved that the system
E
(
ω
,
Z
)
with defect cannot form atomic decomposition for
L
p
(
−
π
,
π
)
,
p
>
1
. The obtained results are the generalizations of those on the atomic decomposition of power weighted exponential system in
L
p
(
−
π
,
π
)
,
p
>
1
, and the frameness of weighted exponential system in
L
2
(
−
π
,
π
)
. |
|---|---|
| AbstractList | This work deals with the frameness of weighted exponential system
E
(
ω
,
Z
)
=
{
ω
(
t
)
e
i
n
t
}
n
∈
Z
in the space
L
p
(
−
π
,
π
)
,
p
>
1
, with the weight function
ω
(
t
)
of general form. Basis properties of
E
(
ω
,
Z
)
in
L
p
(
−
π
,
π
)
,
p
>
1
, are studied, in other words, the criteria of completeness, minimality and basicity of the system
E
(
ω
,
Z
)
in the space
L
p
(
−
π
,
π
)
,
p
>
1
, are given. Sufficient conditions for the completeness and minimality of
E
(
ω
,
Z
∖
F
)
in
L
p
(
−
π
,
π
)
,
p
>
1
, are found, where
F
is an arbitrary finite nonempty subset of the set of integers
Z
. A different method to prove that the system
E
(
ω
,
Z
∖
F
)
does not form a Schauder basis for
L
p
(
−
π
,
π
)
,
p
>
1
, is given. Theorem on a property of expansion system and criterion of Banach frameness for
E
(
ω
,
Z
)
in
L
p
(
−
π
,
π
)
,
p
>
1
, are proved. In particular, it is proved that the system
E
(
ω
,
Z
)
with defect cannot form atomic decomposition for
L
p
(
−
π
,
π
)
,
p
>
1
. The obtained results are the generalizations of those on the atomic decomposition of power weighted exponential system in
L
p
(
−
π
,
π
)
,
p
>
1
, and the frameness of weighted exponential system in
L
2
(
−
π
,
π
)
. This work deals with the frameness of weighted exponential system E(ω,Z)={ω(t)eint}n∈Z in the space Lp(−π,π), p>1, with the weight function ω(t) of general form. Basis properties of E(ω,Z) in Lp(−π,π), p>1, are studied, in other words, the criteria of completeness, minimality and basicity of the system E(ω,Z) in the space Lp(−π,π), p>1, are given. Sufficient conditions for the completeness and minimality of E(ω,Z∖F) in Lp(−π,π), p>1, are found, where F is an arbitrary finite nonempty subset of the set of integers Z. A different method to prove that the system E(ω,Z∖F) does not form a Schauder basis for Lp(−π,π), p>1, is given. Theorem on a property of expansion system and criterion of Banach frameness for E(ω,Z) in Lp(−π,π), p>1, are proved. In particular, it is proved that the system E(ω,Z) with defect cannot form atomic decomposition for Lp(−π,π), p>1. The obtained results are the generalizations of those on the atomic decomposition of power weighted exponential system in Lp(−π,π), p>1, and the frameness of weighted exponential system in L2(−π,π). Abstract This work deals with the frameness of weighted exponential system E ( ω , Z ) = { ω ( t ) e i n t } n ∈ Z $E(\omega ,Z)= \left \{\omega (t)e^{int} \right \}_{n\in Z} $ in the space L p ( − π , π ) $L_{p} (-\pi ,\pi )$ , p > 1 $p>1$ , with the weight function ω ( t ) $\omega (t)$ of general form. Basis properties of E ( ω , Z ) $E(\omega ,Z)$ in L p ( − π , π ) $L_{p} (-\pi ,\pi )$ , p > 1 $p>1$ , are studied, in other words, the criteria of completeness, minimality and basicity of the system E ( ω , Z ) $E(\omega ,Z)$ in the space L p ( − π , π ) $L_{p} (-\pi ,\pi )$ , p > 1 $p>1$ , are given. Sufficient conditions for the completeness and minimality of E ( ω , Z ∖ F ) $E(\omega ,Z\backslash F)$ in L p ( − π , π ) $L_{p} (-\pi ,\pi )$ , p > 1 $p>1$ , are found, where F is an arbitrary finite nonempty subset of the set of integers Z. A different method to prove that the system E ( ω , Z ∖ F ) $E(\omega ,Z\backslash F)$ does not form a Schauder basis for L p ( − π , π ) $L_{p} (-\pi ,\pi )$ , p > 1 $p>1$ , is given. Theorem on a property of expansion system and criterion of Banach frameness for E ( ω , Z ) $E(\omega ,Z)$ in L p ( − π , π ) $L_{p} (-\pi ,\pi )$ , p > 1 $p>1$ , are proved. In particular, it is proved that the system E ( ω , Z ) $E(\omega ,Z)$ with defect cannot form atomic decomposition for L p ( − π , π ) $L_{p} (-\pi ,\pi )$ , p > 1 $p>1$ . The obtained results are the generalizations of those on the atomic decomposition of power weighted exponential system in L p ( − π , π ) $L_{p} (-\pi ,\pi )$ , p > 1 $p>1$ , and the frameness of weighted exponential system in L 2 ( − π , π ) $L_{2} (-\pi ,\pi )$ . |
| ArticleNumber | 26 |
| Author | Simsir Acar, Kader Ismailov, Migdad I. |
| Author_xml | – sequence: 1 givenname: Migdad I. surname: Ismailov fullname: Ismailov, Migdad I. organization: Mathematics, Azerbaijan State Oil and Industry University, Mathematics, Khazar University – sequence: 2 givenname: Kader surname: Simsir Acar fullname: Simsir Acar, Kader email: ksacar@ticaret.edu.tr organization: Institute of Graduate School, İstanbul Ticaret University |
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| Cites_doi | 10.1016/0022-1236(89)90055-4 10.1016/j.jmaa.2005.02.015 10.15672/hujms.1076849 10.1007/BF01321715 10.1007/s10440-011-9663-1 10.1007/s00009-011-0135-7 10.1134/S003744661902006X 10.1090/S0002-9947-1952-0047179-6 10.1007/978-0-8176-4687-5 10.1090/S0002-9947-1973-0312139-8 10.1090/S0002-9904-1974-13458-0 10.4213/im4203 10.1007/s10476-007-0204-0 10.1090/conm/247/03801 |
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| SSID | ssj0002513275 |
| Score | 2.400636 |
| Snippet | This work deals with the frameness of weighted exponential system
E
(
ω
,
Z
)
=
{
ω
(
t
)
e
i
n
t
}
n
∈
Z
in the space
L
p
(
−
π
,
π
)
,
p
>
1
, with the... This work deals with the frameness of weighted exponential system E(ω,Z)={ω(t)eint}n∈Z in the space Lp(−π,π), p>1, with the weight function ω(t) of general... Abstract This work deals with the frameness of weighted exponential system E ( ω , Z ) = { ω ( t ) e i n t } n ∈ Z $E(\omega ,Z)= \left \{\omega (t)e^{int}... |
| SourceID | doaj unpaywall proquest crossref springer |
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| StartPage | 26 |
| SubjectTerms | Analysis Applications of Mathematics Atomic properties Banach spaces Basicity Completeness Decomposition Differential Geometry Frameness Mathematical and Computational Biology Mathematics Mathematics and Statistics Minimality Muckenhoupt condition Topology Weighted exponential system Weighting functions |
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| Title | On Banach frameness of degenerate weighted exponential system |
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