Analysis in Function Spaces Associated with the Group $$ax+b$$

We introduce and describe relations between Sobolev, Besov and Paley–Wiener spaces associated with three representations of the Lie group G of affine transformations of the line, also known as the \( ax + b\) group. 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 K-functional 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.