When Fourier techniques are applied to specific problems from computational finance with nonsmooth functions, the so-called Gibbs phenomenon may become apparent. This seriously affects the efficiency and accuracy of the numerical results. For example, the variance gamma asset price process gives rise to algebraically decaying Fourier coefficients, resulting in a slowly converging Fourier series. We apply spectral filters to achieve faster convergence. Filtering is carried out in Fourier space; the series coefficients are pre-multiplied by a decreasing function. This does not add any significant computational costs. Tests with different filters show how the algebraic index of convergence is improved.