Spectral approximation of convolution operator

We develop a unified framework for constructing matrix approximations to the convolution operator of Volterra type defined by functions that are approximated using classical orthogonal polynomials on $[-1,1]$. The numerically stable algorithms we propose exploit recurrence relations and symmetric properties satisfied by the entries of these convolution matrices. Laguerre-based convolution matrices that approximate Volterra convolution operator defined by functions on $[0,\mathrm{\infty }]$ are also discussed for the sake of completeness.