Let $J_\mu$ denote the Bessel function of order $\mu$. The functions $x^{-\alpha-1} J_{\alpha+2n+1}(x), n=0,1,2,\ldots,$ form an orthogonal system in the space $L^2((0,\infty), x^{2\alpha+1}dx)$ when $\alpha >{-}1$. In this paper we prove that the Fourier series associated to this system is of restricted weak type for the endpoints of the interval of mean convergence, while it is not of weak type if $\alpha\,{\ge}\,0$.