"Gabor frame bound optimization"Shafkulovska, IrinaWe study sharp frame bounds of Gabor systems over rectangular lattices for different windows. In some cases we obtain optimality results for the square lattice while in other cases the lattices optimizing the frame bounds and the condition number are different. Also, in some cases optimal lattices do not exist at all and a degenerated system is optimal. The main results can be summed up as follows: \medskip \begin{center} \fbox{ \parbox{.9\textwidth}{ $\bullet$ \textit{Hyperbolic secant}: the square lattice optimizes the lower and upper frame bound simultaneously and, hence, also the condition number. \medskip $\bullet$ \textit{Cut-off exponentials}: we find that a lattice optimizing the frame bounds does not exist for cut-off exponentials supported on $[0,1/b]$. For support $[0,2/b]$, optimal lattices may or may not exist, depending on the lattice density and which quantity we seek to optimize. If they exist, then they depend on the density and on the decay parameter of the exponential. \medskip $\bullet$ \textit{One-sided exponential}: the optimizing lattices for the condition number and the frame bounds are different from each other and depend on the (over)sampling rate. \medskip $\bullet$ \textit{Two-sided exponentials}: each quantity has a unique optimizer. They depend on the (over)sampling rate and they all differ from each other. } } \end{center} |
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