2007/04/21 by Campos, Rafael G., Mejia, Francisco
#33C45 #44A10 #65D32 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.0704.2842
A discrete Laplace transform and its inversion formula are obtained by using a quadrature of the continuous Fourier transform which is given in terms of Hermite polynomials and its zeros. This approach yields a convergent discrete formula for the two-sided Laplace transform if the function to be transformed falls off rapidly to zero and satisfy certain conditions of integrability, achieving convergence also for singular functions. The inversion formula becomes a quadrature formula for the Bromwich integral. This procedure also yields a quadrature formula for the Mellin transform and its corresponding inversion formula that can be generalized straightforwardly for functions of several variables.