**ABSTRACT**
### Implementations of Shannon's sampling theorem,
a time-frequency approach

#### Thomas Strohmer and Jared Tanner

Shannon's sampling theorem quantifies the Fourier domain periodization
introduced by the equidistant sampling of a bandlimited signal when the
sampling rate is at least as fast as the Nyquist rate dictated by the
signal's bandwidth. If sampled faster than the Nyquist rate,
i.e., oversampling, a reconstruction composed of highly localized atoms is
possible, allowing for practical applications where only a truncated set of
samples is available. More specifically, it is known that root-exponential
accuracy can be achieved by constructing atoms whose Fourier transform
(filter) is infinitely differentiable and compactly supported in the
appropriate bandwidth. Unfortunately, there is no known
compactly supported infinitely smooth filter whose corresponding atom has a
known explicit representation. By considering filters with Gevrey
regularity,
we obtain root-exponential localization for the atom, and an efficient
truncated Gabor approximation of the filter and atom. Furthermore, we
present
an alternative error decomposition that allows for the complete rigorous
analysis of the error in truncating the signal, and of
the error introduced in approximating the filter and atom. By scaling the
approximation order appropriately, the root-exponential convergence rate is
not
adversely effected by the filter's approximation.

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