Given an arbitrary sequence of elements $\xi =\lbrace \xi _n\rbrace _{n\in \mathbb {N}}$ of a Hilbert space $(\mathcal {H},\langle \cdot ,\cdot \rangle )$, the operator $T_\xi$ is defined as the operator associated to the sesquilinear form $\Omega _\xi (f,g)=\sum _{n\in \mathbb {N}} \langle f , \xi _n\rangle \langle \xi _n , g\rangle$, for $f,g\in \lbrace h\in \mathcal {H}: \sum _{n\in \mathbb {N}}|\langle h , \xi _n\rangle |&lt;^&gt;2&lt;\infty \rbrace$. This operator is in general different from the classical frame operator but possesses some remarkable properties. For instance, $T_\xi$ is always self-adjoint with regard to a particular space, unconditionally defined, and, when xi is a lower semiframe, $T_\xi$ gives a simple expression of a dual of xi. The operator $T_\xi$ and lower semiframes are studied in the context of sequences of integer translates of a function of $L^2(\mathbb {R})$. In particular, an explicit expression of $T_\xi$ is given in this context, and a characterization of sequences of integer translates, which are lower semiframes, is proved.

Corso R. (2023). Generalized frame operator, lower semiframes, and sequences of translates. MATHEMATISCHE NACHRICHTEN, 1-19 [10.1002/mana.202000054].

### Generalized frame operator, lower semiframes, and sequences of translates

#### Abstract

Given an arbitrary sequence of elements $\xi =\lbrace \xi _n\rbrace _{n\in \mathbb {N}}$ of a Hilbert space $(\mathcal {H},\langle \cdot ,\cdot \rangle )$, the operator $T_\xi$ is defined as the operator associated to the sesquilinear form $\Omega _\xi (f,g)=\sum _{n\in \mathbb {N}} \langle f , \xi _n\rangle \langle \xi _n , g\rangle$, for $f,g\in \lbrace h\in \mathcal {H}: \sum _{n\in \mathbb {N}}|\langle h , \xi _n\rangle |<^>2<\infty \rbrace$. This operator is in general different from the classical frame operator but possesses some remarkable properties. For instance, $T_\xi$ is always self-adjoint with regard to a particular space, unconditionally defined, and, when xi is a lower semiframe, $T_\xi$ gives a simple expression of a dual of xi. The operator $T_\xi$ and lower semiframes are studied in the context of sequences of integer translates of a function of $L^2(\mathbb {R})$. In particular, an explicit expression of $T_\xi$ is given in this context, and a characterization of sequences of integer translates, which are lower semiframes, is proved.
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Settore MAT/05 - Analisi Matematica
https://onlinelibrary.wiley.com/doi/10.1002/mana.202000054
Corso R. (2023). Generalized frame operator, lower semiframes, and sequences of translates. MATHEMATISCHE NACHRICHTEN, 1-19 [10.1002/mana.202000054].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/588493
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