Real Shannon Wavelet
The Fourier transform of the Shannon mother wavelet is given by:
where the (normalised) gate function is defined by
The analytical expression of the real Shannon wavelet can be found by taking the inverse Fourier transform:
or alternatively as
where
is the usual sinc function that appears in Shannon sampling theorem.
This wavelet belongs to the -class of differentiability, but it decreases slowly at infinity and has no bounded support, since band-limited signals cannot be time-limited.
The scaling function for the Shannon MRA (or Sinc-MRA) is given by the sample function:
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