Interconnection Between the Stretched Exponential Function and the Prony Series: A Concise and Rigorous Revisit

Both the stretched exponential function and the Prony series are prominent mathematical frameworks extensively employed to characterize monotonically decaying phenomena across physics and engineering disciplines. Despite being fundamentally distinct in their mathematical natures, they are profoundly interconnected through elegant theoretical frameworks grounded in Bernstein’s Theorem and the Laplace–Stieltjes transform. This article provides a concise and rigorous revisit that elucidates the theoretical interconnection between the continuous stretched exponential function and the discrete Prony series from two distinct yet complementary perspectives: one rooted in the intuitive approach of the classical calculus framework through the Riemann–Stieltjes integral limit, and the other established upon the contemporary rigor of the measure-theoretic framework through the weak convergence of Borel measures. Regardless of the perspective chosen, we demonstrate that the stretched exponential function represents the exact continuous limit of the Prony series, as the latter theoretically converges to the former as the number of terms approaches infinity. Consequently, the stretched exponential function can be conceptualized as the continuous spectral counterpart of the Prony series, while the latter serves as a discrete representation that formally approaches the structural and topological profile of the former.