  We introduce and describe relations between Sobolev, Besov and Paley-Wiener
spaces associated with three representations of the Lie group of affine
transformations of the line. These representations are left and right regular
representations and a representation in a space of functions defined on the
half-line. The Besov spaces are described as interpolation spaces between
respective Sobolev spaces in terms of the Petree's real interpolation method
and in terms of a relevant moduli of continuity. By using a Laplace operators
associated with these representations a scales of relevant Paley-Wiener spaces
are developed and a corresponding $L_{2}$-approximation theory is constructed
in which our Besov spaces appear as approximation spaces. Another description
of our Besov spaces is given in terms of a frequency-localized Hilbert frames.
A Jackson-type inequalities are also proven.
