An approximation of matrix exponential by a truncated Laguerre series

The Laguerre functions $l_{n,\tau }^{\alpha }$, $n=0,1,\dots$, are constructed from generalized Laguerre polynomials. The functions $l_{n,\tau }^{\alpha }$ depend on two parameters: scale $\tau >0$ and order of generalization $\alpha >-1$, and form an orthogonal basis in $L_2[0,\mathrm{\infty })$. Let the spectrum of a square matrix $A$ lie in the open left half-plane. Then the matrix exponential $H_A(t)=e^{At}$, $t>0$, belongs to $L_2[0,\mathrm{\infty })$. Hence the matrix exponential $H_A$ can be expanded in a series $H_A={\sum }_{n=0}^{\mathrm{\infty }}{S}_{n,\tau ,\alpha ,A}{l}_{n,\tau }^{\alpha }$. An estimate of the norm $\parallel H_A-{\sum }_{n=0}^N{S}_{n,\tau ,\alpha ,A}{l}_{n,\tau }^{\alpha }{\parallel }_{L_2[0,\mathrm{\infty })}$ is proposed. Finding the minimum of this estimate over $\tau$ and $\alpha$ is discussed. Numerical examples show that the optimal $\alpha$ is often almost 0, which essentially simplifies the problem.