Convergence of Padé Kernel Approximants to the Delayed Dirac Function

Abstract

We derive from the solutions of Kummer's equation an expression for the exponential function, which when restricted to integer parameters gives the [M, N] Padé approximation and its error. Laplace inversion, using Bromwich's integral, then yields an approximation of the form

Formula

to δ(λ − t) the delayed Dirac kernel, in which the constants {αi, v, Ki,v: i = 1, 2 , …,N, ν = M − N + 1} are those of the partial fraction decomposition of the [M, N] Padé approximation. These constants also occur in direct quadrature formulae to invert Laplace transforms. Finally, we show that the kernel approximants converge for λ > t.