  By $BMO_o(R)$ we denote the space consisting of all those odd and bounded
mean oscillation functions on R. In this paper we characterize the functions in
$BMO_o(R)$ with bounded support as those ones that can be written as a sum of a
bounded function on $(0,\infty )$ plus the balayage of a Carleson measure on
$(0,\infty )\times (0,\infty )$ with respect to the Poisson semigroup
associated with the Bessel operator $B_\lambda =-x^{-\lambda }Dx^{2\lambda
}Dx^{-\lambda}$, $\lambda >0$. This result can be seen as an extension to
Bessel setting of a classical result due to Carleson.
