On the numerical inversion of the Laplace transform by the use of optimized Legendre polynomials (Q2724923)

For the inversion of the Laplace transform of a real-valued function \( f(t)\), \(t \geq 0 \), in this paper approximations of the form \( \sum_{k=0}^N c_{N,k} e^{-(k \lambda -\beta) t} \) are considered which are obtained by a linear combination of transformed Legendre polynomials. Here \( 0 < \lambda < \beta \) are some parameters, and the coefficients \( c_{N,k} \) depend on a finite number of values of the Laplace transform of \( f \). It is shown that the considered approximations converge uniformly to the function \( f \) on finite intervals if \( f \) belongs to a certain class of functions. That class includes all functions whose second derivative exist and grow at most exponentially as \( t \to \infty \). An optimal choice of the parameters \( \lambda \) and \( \beta \) is considered to minimize the computational complexity, and numerical experiments are presented.