<p>Fourier spectral methods demonstrate an exponential convergence rate when approximating an analytic function. However, for non-smooth and discontinuous functions, this exponential convergence rate diminishes to first order in the smooth regions and produces oscillatory behaviour known as the Gibbs phenomenon. Due to the global nature of spectral methods, these oscillations extend to the whole domain of the function. In our work, we construct a mollifier based on a summability kernel, known as the Rogosinski kernel. Further, we proved that convolving the oscillatory Fourier approximation with the proposed mollifier produces spectral-like convergence. The proof is established by splitting the total error into truncation and regularization errors. Additionally, the parameters are optimised, and the results are numerically compared with those of the existing adaptive Dirichlet mollifier.</p>