  We introduce and study the notions of translation bounded tempered
distributions, and autocorrelation for a tempered distrubution. We further
introduce the spaces of weakly, strongly and null weakly almost periodic
tempered distributions and show that for weakly almost periodic tempered
distributions the Eberlein decomposition holds. For translation bounded
measures all these notions coincide with the classical ones. We show that
tempered distributions with measure Fourier transform are weakly almost
periodic and that for this class, the Eberlein decomposition is exactly the
Fourier dual of the Lesbegue decomposition, with the Fourier-Bohr coefficients
specifying the pure point part of the Fourier transform. We complete the
project by looking at few interesting examples.
