MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINE
For a class of sets with multiple terms $$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times}, \underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots, \underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\},$$having...
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Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics
2020-07-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/209 |
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| author | Elias Zikkos |
| author_facet | Elias Zikkos |
| author_sort | Elias Zikkos |
| collection | DOAJ |
| description | For a class of sets with multiple terms
$$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times},
\underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots,
\underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\},$$having density \(d\) counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers\linebr eak \(\{d_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots ,\mu_n-1\} \) satisfying certain growth conditions, we consider a moment problem of the form $$\int_{-\infty}^{\infty}e^{-2w(t)}t^k e^{\lambda_n t}f(t)\, dt=d_{n,k},\quad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,2,\dots, \mu_n-1,$$ in weighted \(L^2 (-\infty, \infty)\) spaces. We obtain a solution \(f\) which extends analytically as an entire function, admitting a Taylor–Dirichlet series representation $$ f(z)=\sum_{n=1}^{\infty}\Big(\sum_{k=0}^{\mu_n-1}c_{n,k} z^k\Big) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C},\quad\forall\,\, z\in \mathbb{C}. $$ The proof depends on our previous work where we characterized the closed span of the exponential system \(\{t^k e^{\lambda_n t}:\, n\in\mathbb{N},\,\, k=0,1,2,\dots,\mu_n-1\}\) in weighted \(L^2 (-\infty, \infty)\) spaces, and also derived a sharp upper bound for the norm of elements of a biorthogonal sequence to the exponential system. The proof also utilizes notions from Non-Harmonic Fourier series such as Bessel and Riesz–Fischer sequences. |
| format | Article |
| id | doaj-art-3e3f1a0d0e7f44cea3eefd066b5ac66e |
| institution | Matheson Library |
| issn | 2414-3952 |
| language | English |
| publishDate | 2020-07-01 |
| publisher | Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj-art-3e3f1a0d0e7f44cea3eefd066b5ac66e2025-08-02T05:57:57ZengUral Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and MechanicsUral Mathematical Journal2414-39522020-07-016110.15826/umj.2020.1.014100MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINEElias Zikkos0Khalifa University, PO. Box 127788 Abu DhabiFor a class of sets with multiple terms $$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times}, \underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots, \underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\},$$having density \(d\) counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers\linebr eak \(\{d_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots ,\mu_n-1\} \) satisfying certain growth conditions, we consider a moment problem of the form $$\int_{-\infty}^{\infty}e^{-2w(t)}t^k e^{\lambda_n t}f(t)\, dt=d_{n,k},\quad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,2,\dots, \mu_n-1,$$ in weighted \(L^2 (-\infty, \infty)\) spaces. We obtain a solution \(f\) which extends analytically as an entire function, admitting a Taylor–Dirichlet series representation $$ f(z)=\sum_{n=1}^{\infty}\Big(\sum_{k=0}^{\mu_n-1}c_{n,k} z^k\Big) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C},\quad\forall\,\, z\in \mathbb{C}. $$ The proof depends on our previous work where we characterized the closed span of the exponential system \(\{t^k e^{\lambda_n t}:\, n\in\mathbb{N},\,\, k=0,1,2,\dots,\mu_n-1\}\) in weighted \(L^2 (-\infty, \infty)\) spaces, and also derived a sharp upper bound for the norm of elements of a biorthogonal sequence to the exponential system. The proof also utilizes notions from Non-Harmonic Fourier series such as Bessel and Riesz–Fischer sequences.https://umjuran.ru/index.php/umj/article/view/209moment problems, exponential systems, biorthogonal families, weighted banach spaces, bessel and riesz–fischer sequences |
| spellingShingle | Elias Zikkos MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINE Ural Mathematical Journal moment problems, exponential systems, biorthogonal families, weighted banach spaces, bessel and riesz–fischer sequences |
| title | MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINE |
| title_full | MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINE |
| title_fullStr | MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINE |
| title_full_unstemmed | MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINE |
| title_short | MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINE |
| title_sort | moment problems in weighted l 2 spaces on the real line |
| topic | moment problems, exponential systems, biorthogonal families, weighted banach spaces, bessel and riesz–fischer sequences |
| url | https://umjuran.ru/index.php/umj/article/view/209 |
| work_keys_str_mv | AT eliaszikkos momentproblemsinweightedl2spacesontherealline |