  A fast non-polynomial interpolation is proposed in this paper for functions
with logarithmic singularities. It can be executed fast with the discrete
cosine transform. Based on this interpolation, a new quadrature is proposed for
a kind of logarithmically singular integrals. The interpolation and integration
errors are also analyzed. Numerical examples of the interpolation and
integration are shown to validate the efficiency of the proposed new
interpolation and the new quadrature.
