We prove the failure of a.e. convergence of the Fourier expansion in terms of the orthonormal polynomials with respect to the measure $(1-x)^\alpha(1+x)^\beta dx+M\delta_{-1}+N\delta_1$, where $\delta_t$ is the delta function at a point $t$ and $M>0$, $N>0$. Lebesgue norms of Koornwinder’s Jacobi-type polynomials are applied to obtain a new proof of necessary conditions for mean convergence.