Naslov
The coefficient space of systems of integer translates

Autori
Berić, Tomislav

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, kratko priopćenje, znanstveni

Skup
International Workshop on Frames, Wavelets, Approximation Methods and Applications

Mjesto i datum
Palermo, Italija, 16.09.2019. - 19.09.2019

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Nije recenziran

Ključne riječi
Shift invariant systems, Bases, Frames, Riesz bases, Periodization function, Besselian property, Hilbertian property

Sažetak
We denote by $T_a : L^2(\R) \to L^2(\R)$ the translation operator defined as $(T_a f)(x) := f(x - a)$. For a given function $\psi \in L^2(\R)$ we will study the system of integer translates $B_\psi = \left\{; ; T_k \psi : k \in \Z \right\}; ; $. Numerous properties of $B_\psi$ can be described via its periodization function $p_\psi(\xi) = \sum_{; ; k \in \Z}; ; \left| \widehat{; ; \psi}; ; (\xi + k) \right|^2$. For $\psi$ we define its associated coefficient space $\Cof_\psi$ as the set of all the sequences $\left( c_k \right)_{; ; k \in \Z}; ; $ for which $\sum c_k T_k \psi$ converges in the $L^2$ norm. The coefficient space depends implicitly on the enumeration of $\Z$ and here we use the standard enumeration: $0, -1, 1, -2, 2, \ldots$ There are two important special cases: when $\Cof_\psi$ contains $\ell^2(\Z)$, in which case we say that $B_\psi$ has the $(H)$-- property, and when $\Cof_\psi$ is contained in $\ell^2(\Z)$, when we say that $B_\psi$ has the $(B)$--property. It is the latter case we devote the most attention to since the former case has already been studied and characterized via the periodization function in \cite{; ; HSWW}; ; . This is joint work with Hrvoje Šikić. \begin{; ; thebibliography}; ; {; ; 00}; ; \bibitem{; ; HSWW}; ; E.~Hern\'{; ; a}; ; ndez, H.~\v{; ; S}; ; iki\'{; ; c}; ; , G.~Weiss, E.~Wilson, \emph{; ; On the properties of the integer translates of a square integrable function}; ; , Harmonic analysis and partial differential equations, Contemp. Math., vol. 505, Amer. Math. Soc., Providence, RI, 2010, pp. 233--249. MR 2664571 (2011g:42077) \end{; ; thebibliography}; ;

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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