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

Abstract

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.

1. Introduction

Both the stretched exponential function and the Prony series are categorized as completely monotonic functions [1,2,3,4,5,6,7,8,9,10,11,12,13,14], exhibiting strictly decreasing, convex profiles on the positive real axis; that is, the function $f\left(t\right)$, representing either case, satisfies ${\left(-1\right)}^n{f}^{\left(n\right)}\left(t\right)\ge 0$ for all $n\in {\mathbb{N}}_0$ and $t\in \left(0,\infty \right)$ (i.e., $f\left(t\right)\ge 0$, $f^{\prime }\left(t\right)\le 0$, $f^{″}\left(t\right)\ge 0$, $f^{‴}\left(t\right)\le 0$, etc.) [2,4,12,15,16,17,18]. Consequently, both functions are prominent mathematical frameworks for characterizing monotonic decay across many physics and engineering disciplines, including stress relaxation in viscoelasticity [19,20,21,22,23,24,25,26], dielectric relaxation [5,27,28,29,30,31,32,33], luminescence lifetime decay [6,34,35,36,37,38,39], magnetic relaxation [40,41,42], and non-ideal discharge of capacitors [43,44,45,46].

The complete monotonicity of both the stretched exponential function and the Prony series ensures physically meaningful representations, thereby guaranteeing non-negative spectra, which are essential for modeling physically consistent and admissible behaviors; their non-negativity implies that energy dissipation cannot be negative, which satisfies the non-negative dissipation requirement of the second law of thermodynamics and ensures the physical consistency and admissibility of the model [5,10,47,48,49,50,51]. From the physical perspective, ensuring positive weighting coefficients for the stretched exponential function and the Prony series in relaxation phenomena guarantees a completely monotonic relaxation function. This condition ensures the non-negativity of the underlying relaxation spectra, thereby precluding unphysical negative energy dissipation [3,5,52]. In contrast, models with negative coefficients can result in non-monotonic relaxation, no energy dissipation or even energy gain, which are typically associated with active metamaterials rather than standard passive media [52].

Despite their shared use in modeling monotonic decay, the stretched exponential function and the Prony series are fundamentally distinct in their mathematical natures. The former is a continuous function designed to represent a continuous distribution of relaxation processes to capture complex, multi-scale dynamics. In contrast, the latter provides a discrete representation that approximates such behavior using a finite sum of exponential basis functions. Nevertheless, they are profoundly interconnected [6,11,26,31,38,53,54,55,56,57,58] through elegant theoretical frameworks grounded in Bernstein’s Theorem and the Laplace–Stieltjes transform.

The purpose of this article is to examine the theoretical interconnection between the stretched exponential function and the Prony series, while elucidating the underlying theoretical foundations. The remainder of this article is structured as follows. Section 2 outlines the stretched exponential function and the Prony series, covering their definitions, formulations, and key fundamental concepts. Section 3 delves into the interconnection between these two functions from two distinct, yet complementary, theoretical 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. This is followed by the Discussion and Conclusions sections. This article aims to provide a concise yet rigorous resource that facilitates a comprehensive understanding of the interconnection between the stretched exponential function and the Prony series, while elucidating the theoretical foundations that underpin this interconnection.

2. Overview of Stretched Exponential Function and Prony Series

2.1. Stretched Exponential Function

The stretched exponential function, also known as the Kohlrausch–Williams–Watts (KWW) function [59,60,61,62,63] or the complementary cumulative Weibull function [64,65], is a real-valued function defined for $t\in \left[\left.0,\infty \right)\right.$, such that

$$\varphi \left(t\right)={\varphi }_0{e}^{-{\left({\displaystyle \frac{t}{{\tau }_0}}\right)}^{\beta }},$$

(1)

where ${\varphi }_0$ is the initial value at $t=0$, ${\tau }_0\in \left(0,\infty \right)$ is the characteristic relaxation time (or scale parameter), and $\beta \in \left(\left.0,1\right]\right.$ is the stretching exponent (or shape parameter). For the sake of mathematical clarity and without loss of generality, we consider the normalized form of the stretched exponential function where ${\varphi }_0=1$ hereafter in this article.

The stretched exponential function was originally introduced by Rudolf Kohlrausch in 1854 to describe the anomalous, non-exponential discharge of a Leyden jar (capacitor) [66], and it was latter used by Williams and Watts in 1970s to describe dielectric relaxation [67,68], thereby giving rise to the so-called Kohlrausch–Williams–Watts (KWW) function.

The stretched exponential function can be interpreted either as a generalization of the standard exponential decay through the introduction of a fractional power law in the argument, or as a synthesis of standard exponential and power-law behaviors.

The behavior of the stretched exponential function is sensitive to the value of $\beta$, exhibiting fundamental changes that can be examined across the following three scenarios:

(1)

$\beta =1$: The stretched exponential function reduces to the standard Debye exponential decay.

(2)

$\beta <1$, as $\beta$ approaches 1: The stretched exponential function converges to the standard Debye exponential decay.

(3)

$\beta <1$, as $\beta$ approaches 0: The stretched exponential function exhibits a more pronounced divergence from the standard Debye exponential decay. Specifically, a lower value of $\beta$ manifests as a more rapid initial drop followed by a slower, long-tailed relaxation process. Mathematically, this behavior corresponds to a broader continuous relaxation spectrum (i.e., the distribution of relaxation times becomes increasingly broad). Physically, a low value of $\beta$ is often associated with material heterogeneity or a higher degree of structural disorder within the system [69,70,71,72,73].

2.2. Prony Series

The Prony series, a specific type of the Dirichlet series commonly defined as a finite linear combination of exponential decay functions, is a real-valued function defined for $t\in \left[\left.0,\infty \right)\right.$, such that

$$g\left(t\right)={\displaystyle \sum _{i=1}^N}{g}_i{e}^{-{\displaystyle \frac{t}{{\tau }_i}}},$$

(2)

where $N\in \mathbb{N}$ is the order (also referred to as the number of terms or Prony modes) of the series, $g_i\in \left[\left.0,\infty \right)\right.$ is the weighting coefficient (or amplitude) of the $i$-th term that governs its contribution to the overall response, and ${\tau }_i\in \left(0,\infty \right)$ is the characteristic relaxation time that governs the decay rate of the $i$-th term. For the sake of mathematical clarity and without loss of generality, we consider the normalized form of the Prony series where the dimensionless weighting coefficients $g_i$ satisfying ${\displaystyle {\sum }_{i=1}^N}{g}_i=1$ hereafter in this article.

The Prony series was originally introduced by French engineer and mathematician Gaspard Riche de Prony in 1795 [74], and has become a standard approach since the twentieth century to describe the time-dependent mechanical behaviors of viscoelastic materials.

From the perspective of functional analysis, the Prony series serves as a discrete approximation of a completely monotonic function. In the limit as $N\to \infty$, this discrete sum theoretically converges to the underlying continuous spectral representation. This transition provides the conceptual cornerstone for the interconnection between the Prony series and the stretched exponential function, a topic that will be explored in the subsequent section.

3. Interconnection Between Stretched Exponential Function and Prony Series

The fundamental link between the stretched exponential function and the Prony series is that the stretched exponential function represents the continuous limit of the Prony series as the latter theoretically converges to the former as the number of terms approaches infinity ($N\to \infty$). It can thus be stated that the stretched exponential function represents a 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, i.e.,

$$e^{-{\left({\displaystyle \frac{t}{{\tau }_0}}\right)}^{\beta }}\approx {\displaystyle \sum _{i=1}^N}{g}_i{e}^{-{\displaystyle \frac{t}{{\tau }_i}}},\mathrm{a}\mathrm{s}N\to \infty .$$

(3)

In the following, we examine the interconnection between the stretched exponential function and the Prony series from two distinct, yet complementary, theoretical perspectives. Section 3.1 demonstrates the convergence of the discrete Prony series to the continuous stretched exponential function through the classical calculus framework of a Riemann–Stieltjes integral limit over an absolutely continuous spectral representation, with the underlying mathematical significance and structural insights discussed at the end of the section. Section 3.2 formalizes this interconnection from a contemporary measure-theoretic framework, establishing a unified discrete-continuous spectral relationship between these two representations through the weak convergence of Borel measures.

3.1. Convergence of Prony Series to Stretched Exponential Function

Consider the Prony series represented as a discrete sum of exponential decay functions:

$$g\left(t\right)={\displaystyle \sum _{i=1}^N}{g}_i{e}^{-{\displaystyle \frac{t}{{\tau }_i}}}.$$

(4)

To bridge the discrete and continuous representations, we express the discrete weighting coefficients $g_i$ at $\tau ={\tau }_i$ through an underlying continuous spectral density $H\left(\tau \right)$. Specifically, by localizing this continuous spectral density over finite intervals $∆{\tau }_i$, we write $g_i\left({\tau }_i\right)=H\left({\tau }_i\right)∆{\tau }_i$. Unlike the discrete weighting coefficients $g_i$, which represent point intensities, $H\left(\tau \right)$ defines the spectral intensity per unit interval of $\tau$. From the classical and pedagogical perspective, this step conceptualizes the Prony series as a formal discrete representation that mimics the structure of a continuous spectral density $H\left(\tau \right)$ over a partitioned domain. Substituting $g_i=H\left({\tau }_i\right)∆{\tau }_i$ into Equation (4) yields

$$g\left(t\right)={\displaystyle \sum _{i=1}^N}H\left({\tau }_i\right)e^{-{\displaystyle \frac{t}{{\tau }_i}}}∆{\tau }_i.$$

(5)

From the classical perspective, Equation (5) can be conceptualized as a formal Riemann–Stieltjes sum for the kernel $H\left(\tau \right)e^{-{\displaystyle \frac{t}{\tau }}}$ over a partitioned domain. Under this framework, the transition behaves analogously to an idealized discrete quadrature sequence approaching the continuous Riemann integral representation as the partition becomes infinitely fine ($∆{\tau }_i\to 0$ as $N\to \infty$):

$$\underset{N\to \infty }{\lim }{\displaystyle \sum _{i=1}^N}H\left({\tau }_i\right)e^{-{\displaystyle \frac{t}{{\tau }_i}}}∆{\tau }_i={\int }_0^{\infty }H\left(\tau \right)e^{-{\displaystyle \frac{t}{\tau }}}d\tau .$$

(6)

The integral in Equation (6) constitutes the Laplace–Stieltjes transform of a spectral distribution function $\mu \left(\tau \right)$ such that ${\displaystyle \frac{d\mu \left(\tau \right)}{d\tau }}=H\left(\tau \right)$. To rigorously establish the existence of a continuous spectral density for the stretched exponential function, we invoke the following classical theorem:

Lemma 1

(Bernstein’s Theorem on completely monotonic functions) [75]. A real-valued function $f\left(t\right)$ defined on $\left[\left.0,\infty \right)\right.$ is completely monotonic on $\left(0,\infty \right)$ if and only if it can be represented as the Laplace–Stieltjes transform of a non-negative, bounded Borel measure $\mu$ on $\left[\left.0,\infty \right)\right.$, such that

$$f\left(t\right)={\int }_0^{\infty }{e}^{-{\displaystyle \frac{t}{\tau }}}d\mu \left(\tau \right),$$

for all $t>0$.

By applying Bernstein’s Theorem to the stretched exponential function $e^{-{\left({\displaystyle \frac{t}{{\tau }_0}}\right)}^{\beta }}$, its complete monotonicity guarantees the existence of a unique spectral measure. By assuming the absolute continuity of this measure under the classical integral framework, there must exist a non-negative continuous spectral density $H\left(\tau \right)$ such that Equation (6) equates to the stretched exponential function:

$$e^{-{\left({\displaystyle \frac{t}{{\tau }_0}}\right)}^{\beta }}={\int }_0^{\infty }H\left(\tau \right)e^{-{\displaystyle \frac{t}{\tau }}}d\tau .$$

(7)

Since $H\left(\tau \right)$ is inherently dependent on $\beta$, Equation (7) is more precisely rewritten as

$$e^{-{\left({\displaystyle \frac{t}{{\tau }_0}}\right)}^{\beta }}={\int }_0^{\infty }H\left(\tau ,\beta \right)e^{-{\displaystyle \frac{t}{\tau }}}d\tau .$$

(8)

By performing the inverse Laplace transform on Equation (8), $H\left(\tau ,\beta \right)$ can be obtained explicitly as

$$H\left(\tau ,\beta \right)={\displaystyle \frac{{\tau }_0}{\pi {\tau }^2}}{\int }_0^{\infty }{e}^{-{\displaystyle \frac{{\tau }_{0}u}{\tau }}}{e}^{-u^{\beta }\cos \left(\pi \beta \right)}\sin \left(u^{\beta }\sin \left(\pi \beta \right)\right)du.$$

(9)

This integral representation, first obtained by Pollard [1,2,35,76], is the one-sided Lévy stable distribution. This completes the demonstration of the convergence of the Prony series to the stretched exponential function as $N\to \infty$ through the classical calculus framework.

Regarding Lemma 1, we must emphasize a critical issue concerning the domains of complete monotonicity for both discrete and continuous representations. The discrete Prony series $g\left(t\right)={\displaystyle {\sum }_{i=1}^N}{g}_i{e}^{-{\displaystyle \frac{t}{{\tau }_i}}}$ is infinitely differentiable at $t=0$, meaning its complete monotonicity holds on the closed interval $\left[\left.0,\infty \right)\right.$. However, the continuous stretched exponential function $\varphi \left(t\right)=e^{-{\left({\displaystyle \frac{t}{{\tau }_0}}\right)}^{\beta }}$ is completely monotonic only on the open interval $\left(0,\infty \right)$, excluding the initial point $t=0$; this restriction arises because as $t\to 0^{+}$, its first derivative ${\varphi }^{\prime }\left(t\right)\propto t^{\beta -1}$ approaches $-\infty$, rendering all higher-order derivatives singular at $t=0$. This boundary singularity at $t=0$ profoundly impacts both the continuous spectral density $H\left(\tau ,\beta \right)$ and the convergence topology of the discrete Prony series approximation near $t=0$. On one hand, to physically compensate for the infinite relaxation rate (${\varphi }^{\prime }\left(t\right)\to -\infty$) at the instantaneous onset of relaxation, the underlying continuous spectral density $H\left(\tau ,\beta \right)$ must exhibit severe power-law divergence at the short relaxation-time limit ($\tau \to 0$), representing an infinite concentration of fast-relaxing microscopic processes. One the other hand, this asymmetry severely impairs the convergence behavior of the discrete Prony series near $t=0$. Since any finite truncation ${\displaystyle {\sum }_{i=1}^N}{g}_i{e}^{-{\displaystyle \frac{t}{{\tau }_i}}}$ possesses strictly bounded derivatives at $t=0$, employing this regular analytic function to approximate a function with singularity at $t=0$ results in non-uniform convergence. Consequently, a standard discrete Prony series approximation exhibits pronounced fitting stagnation or oscillations near $t=0$, demanding an exponentially dense clustering of discrete relaxation times ${\tau }_i$ near $t=0$ to capture the initial singular decay slope.

The mathematical significance and structural insights of the convergence of the Prony series to the stretched exponential function, which extends beyond the standard transition from a Riemann–Stieltjes sum to integral, are discussed below:

(1)

From the perspective of approximation theory and computational inverse problems, the discrete Prony series is inherently susceptible to severe ill-posedness. Since the exponential basis functions $e^{-{\displaystyle \frac{t}{{\tau }_i}}}$ become nearly collinear when the relaxation times ${\tau }_i$ are densely distributed, the resulting kernel matrix exhibits an exponentially growing condition number. Consequently, optimization algorithms often fail to yield unique or stable solutions for the weights $g_i$ and relaxation times ${\tau }_i$ under noisy data. In practical applications, transforming a continuous spectral representation back into a stable discrete counterpart constitutes a Fredholm integral equation of the first kind. To resolve this ill-posedness, advanced regularized inversion techniques must be deployed, such as Tikhonov regularization (which introduces an ${\mathcal{L}}_2$ norm penalty to smooth out non-uniqueness) or maximum entropy methods (which enforce physical non-negativity and informational constraints to select optimal discrete clusters of $g_i$ and ${\tau }_i$). However, the convergence to the continuous integral in Equation (8) fundamentally alters the topology of the parameter space. By transitioning from a discrete point spectrum to a continuous spectral density $H\left(\tau ,\beta \right)$, the continuous representation smooths out numerical instabilities, yielding a stable and analytical distribution function. In this sense, the continuous stretched exponential function can be interpreted as a natural regularization that analytically bypasses the numerical instability of the discrete Prony series.

(2)

This convergence clarifies how macroscopic multi-scale, non-local dynamics spontaneously arise from single-scale, local differential operators. Each individual term in the finite Prony series satisfies a first-order linear differential equation, representing a single-scale, local memoryless Markovian relaxation process. However, the limiting continuous integral representation demonstrates that the stretched exponential function manifests as an infinite, weighted superposition of these elementary processes. Physically, this uncovers a profound emergent phenomenon: the macroscopic memory effect (non-locality) emerges collectively from an infinite ensemble of microscopic, elementary components that are inherently memoryless (local). Consequently, this dual representation establishes a rigorous analytical bridge linking single-scale, local, discrete and multi-scale, non-local, continuous formulations.

3.2. Unified Discrete-Continuous Spectral Interconnection Between Prony Series and Stretched Exponential Function Through the Weak Convergence of Borel Measures

In this section, we construct the interconnection between the discrete Prony series and the continuous stretched exponential function from a contemporary measure-theoretic framework, establishing a unified discrete-continuous spectral interconnection between these two representations through the weak convergence of Borel measures.

By virtue of Bernstein’s Theorem, the complete monotonicity of the stretched exponential function guarantees the existence of a unique, non-negative, bounded continuous Borel measure $\mu$ on the Borel $\sigma$-algebra $\mathcal{B}\left({\mathbb{R}}_{>0}\right)$, such that

$$e^{-{\left({\displaystyle \frac{t}{{\tau }_0}}\right)}^{\beta }}={\int }_0^{\infty }{e}^{-{\displaystyle \frac{t}{\tau }}}d\mu \left(\tau \right),$$

(10)

where $\mu$ is Lebesgue-absolutely continuous, admitting the smooth spectral density $H\left(\tau ,\beta \right)={\displaystyle \frac{d\mu \left(\tau \right)}{d\tau }}$ derived in Equation (9). This integral formally defines the continuous spectral topology of the system. To establish a discrete spectral counterpart within the same measure-theoretic framework, we construct a discrete (atomic) Borel measure ${\mu }_N$ supported on a finite, appropriately selected set of distinct relaxation time ${\left\{{\tau }_i\right\}}_{i=1}^N\subset \left(0,\infty \right)$:

$${\mu }_N={\displaystyle \sum _{i=1}^N}{g}_i{\delta }_{{\tau }_i},$$

(11)

where ${\delta }_{{\tau }_i}$ denotes the Dirac measure (or point mass) centered at ${\tau }_i$, and $g_i>0$ denotes the discrete weighting coefficients satisfying ${\displaystyle {\sum }_{i=1}^N}{g}_i=1$. By evaluating the Lebesgue-Stieltjes integral of the exponential kernel $e^{-{\displaystyle \frac{t}{\tau }}}$ against this discrete measure ${\mu }_N$, and invoking the linearity of integral along with the sifting property of the Dirac measure (where ${\int }_0^{\infty }\psi \left(\tau \right)d{\delta }_{{\tau }_i}\left(\tau \right)=\psi \left({\tau }_i\right)$), the integral proceeds explicitly as follows:

$${\int }_0^{\infty }{e}^{-{\displaystyle \frac{t}{\tau }}}d{\mu }_N\left(\tau \right)={\int }_0^{\infty }{e}^{-{\displaystyle \frac{t}{\tau }}}d\left({\displaystyle \sum _{i=1}^N}{g}_i{\delta }_{{\tau }_i}\left(\tau \right)\right)={\displaystyle \sum _{i=1}^N}{g}_i{\int }_0^{\infty }{e}^{-{\displaystyle \frac{t}{\tau }}}d{\delta }_{{\tau }_i}\left(\tau \right)={\displaystyle \sum _{i=1}^N}{g}_i{e}^{-{\displaystyle \frac{t}{{\tau }_i}}}.$$

(12)

Equation (12) demonstrates that the Prony series is the exact, rigorous integral representation of the discrete measure ${\mu }_N$. The core mathematical interconnection between the discrete Prony series and the continuous stretched exponential function lies in the topological concept of weak convergence (denoted as ${\mu }_N\rightharpoonup \mu$). In measure theory, a discrete measure ${\mu }_N$ is said to converge weakly to a continuous measure $\mu$ if, for any continuous and bounded test function $\psi \left(\tau \right)$ on ${\mathbb{R}}_{>0}$, the following limit holds:

$$\underset{N⟶\infty }{\lim }{\int }_0^{\infty }\psi \left(\tau \right)d{\mu }_N\left(\tau \right)={\int }_0^{\infty }\psi \left(\tau \right)d\mu \left(\tau \right).$$

(13)

Since the kernel $\psi \left(\tau \right)=e^{-{\displaystyle \frac{t}{\tau }}}$ is strictly continuous and bounded for all $t\ge 0$ and $\tau \in \left(0,\infty \right)$, it serves as a test function. Substituting $\psi \left(\tau \right)=e^{-{\displaystyle \frac{t}{\tau }}}$, Equation (10) and Equation (12) into the weak convergence definition of Equation (13) yields:

$$\underset{N⟶\infty }{\lim }{\displaystyle \sum _{i=1}^N}{g}_i{e}^{-{\displaystyle \frac{t}{{\tau }_i}}}={\int }_0^{\infty }{e}^{-{\displaystyle \frac{t}{\tau }}}d\mu \left(\tau \right)=e^{-{\left({\displaystyle \frac{t}{{\tau }_0}}\right)}^{\beta }}.$$

(14)

This completes the demonstration that the stretched exponential function represents the continuous limit of the Prony series, as the latter theoretically converges to the former as the number of terms approaches infinity ($N\to \infty$).

It is critical to acknowledge that Equation (14) represents a non-constructive topological approximation rather than an explicit algebraic expansion. This formulation does not provide a direct algorithm for extracting the specific parameters $g_i$ and ${\tau }_i$ for a finite truncated $N$. Instead, it rigorously demonstrates that the Prony series serves as a valid discrete measure that weakly traces the continuous topology of the underlying spectral density of the stretched exponential function, thereby establishing a rigorous and elegant foundation that unifies the discrete and continuous representations.

4. Discussion

In this article, we revisit the interconnection between the stretched exponential function and the Prony series from two distinct, yet complementary, theoretical perspectives, and demonstrate that Bernstein’s Theorem and the Laplace–Stieltjes transform serve as two theoretical underpinnings that fundamentally bridge these two prominent mathematical frameworks for characterizing monotonic decay across physics and engineering disciplines.

The mathematical bridge between the stretched exponential function and the Prony series is anchored by Bernstein’s Theorem, which unifies both under the umbrella of completely monotonic functions that rigorously admit Laplace–Stieltjes transforms of non-negative measures. The transition from the discrete Prony series to the continuous stretched exponential function is governed by this shared transform, with the former acting as a discrete representation that formally approaches the latter’s underlying continuous spectral density. In the limit as the number of terms of the Prony series approaches infinity, its discrete sum converges to the continuous integral representation, ultimately recovering the stretched exponential function.

Conceptually, the Prony series can be viewed as a discrete, finite sampling of the continuous spectral density. In contrast, the stretched exponential function represents an idealized continuous-spectrum function in the limit of an infinitely dense distribution of sampling points. In this perspective, the Prony series serves as a discretized manifestation, fragmenting a continuous natural process into a set of discrete components for numerical tractability. Consequently, the distinction between these two representations lies in their spectral topology: the Prony series and the stretched exponential function are characterized by discrete and continuous spectra, respectively, implying that they are fundamentally distinct manifestations of the same mathematical entity. Physically, both representations characterize the same dissipative energy processes, manifested as a weighted superposition of relaxation mechanisms across multiple timescales.

The spectral representations of both the stretched exponential function and the Prony series can be formally unified through the framework of Borel measures. Invoking Berstein’s Theorem, the complete monotonicity of these two functions ensures their representations as Laplace–Stieltjes transforms of non-negative measures. The distinction between the two functions lies in the nature of their underlying measures: the Prony series is characterized by a discrete measure, comprising a linear combination of Dirac deltas with weights $g_i$. In contrast, the stretched exponential function is characterized by a continuous measure, defined by its continuous spectral density. In the limit, as the number of terms of the Prony series approaches infinity and the sampling of relaxation times becomes infinitely dense, the discrete distribution of weights $g_i$ converges to the continuous spectral density. Physically, these measures represent the distributional sources of energy dissipation. The mathematical requirement of non-negativity for these measures, a prerequisite for complete monotonicity, is fundamentally important. It ensures strict non-negativity of energy dissipation, guaranteeing that the model complies with the second law of thermodynamics by precluding the spontaneous generation of energy.

Despite both the stretched exponential function and the Prony series are prominent mathematical frameworks for characterizing monotonic decay, they possess distinct characteristics from the computational and practical perspectives, beyond their aforementioned mathematical distinctions. The primary advantage of the stretched exponential function lies in its parsimony, as it can successfully fit a wide range of complex, multi-scale monotonically decaying phenomena, serving as a universal law for characterizing those phenomena. However, this parsimonious form introduces several computational and practical challenges, many of which are effectively addressed by the Prony series. The trade-offs between these two functions can be examined across the following four key points.

(1)

Computational cost and complexity: The stretched exponential function involves a non-integer power ${\left({\displaystyle \frac{t}{{\tau }_0}}\right)}^{\beta }$. Consequently, it lacks efficient recursive update algorithms for the numerical implementation of the stretched exponential function. In contrast, the Prony series, as a sum of simple exponential decays, is exceptionally efficient for recursive update algorithms, which is why it remains the default model for viscoelasticity in commercial finite element analysis software.

(2)

Spectral density accessibility: Deriving the analytical closed-form solution of the underlying relaxation time distribution from the stretched exponential function is mathematically cumbersome. Except for special cases (e.g., $\beta =0.5$) [2], its spectral density $H\left(\tau ,\beta \right)$ requires complex series expansion or numerical integration. However, the Prony series provides a direct and intuitive spectral representation through its discrete weighting coefficients $g_i$.

(3)

Frequency domain conversion: In applications such as dynamic mechanical analysis in viscoelasticity, the Fourier transform of the stretched exponential functions lacks an analytical closed-form solution, complicating the analysis of frequency-dependent responses. The Prony series provides a significant advantage in this regard, as its Fourier transform yields straightforward algebraic expressions for frequency-dependent responses such as storage and loss moduli.

(4)

Construction of equivalent circuit models: The stretched exponential function lacks a direct correspondence to a combination of a finite number of classical mechanical or electrical elements, which complicates the construction of equivalent circuit models. In contrast, each term in the Prony series corresponds to a fundamental spring-dashpot assembly or an equivalent resistance-capacitance circuit. Consequently, the Prony series is uniquely suited for constructing equivalent circuit models. This compatibility allows for integration into simulation environments such as SPICE, enabling efficient modeling of various monotonically decaying phenomena at the circuit level.

Based on the arguments presented in the preceding paragraph, from the theoretical and physical perspectives, the stretched exponential function remains the superior choice for its elegance and capacity to serve as a universal law for characterizing monotonically decaying phenomena, but may be less convenient numerically. In contrast, the Prony series serves as an indispensable tool for numerical implementation, bridging the gap between idealized physical laws and practical numerical implementation [21,25,77]. Nevertheless, it is important to note that the Prony series may require an extensive number of terms to accurately fit highly non-exponential decays, leading to parametric redundancy. Furthermore, it is often susceptible to ill-posedness; due to the inherent functional similarity among its exponential basis functions, the optimization process often suffers from numerical instability, resulting in a lack of unique fitting results. In contrast, the stretched exponential function inherently circumvents these two challenges by providing a parsimonious functional form with a minimal parameter set.

The present work, while providing a rigorous theoretical foundation for the discrete-continuous spectral transition, lays a solid foundation for subsequent advanced computational studies. To illustrate this, a few examples are presented below. First, we can select a specific stretched exponential function (e.g., setting parameters to $\beta =0.5$ and ${\tau }_0=1$), analytically compute its continuous spectrum $H\left(\tau ,\beta \right)$ using Pollard’s formula, and numerically construct the $N$-term Prony series under various collocation or sampling densities. Second, we can provide a plot illustrating the relative $L_2$ or $L_{\infty }$ error between a specific stretched exponential function and its Prony series approximation as a function of the number of terms $N$; this result empirically demonstrates how the discrete quadrature converges to the continuous integral representation as $N\to \infty$. Third, we can investigate the behavior of the Prony series approximation to the stretched exponential function in extreme physical regimes, such as for $\beta <0.3$, where the continuous spectral density features a sharp singularity near the origin. These investigations will not only empirically validate the established theoretical convergence but also provide the practical utility required for high-precision modeling in viscoelasticity, rheology, materials science and engineering, and a wide range of complex systems and applied mathematical problems [78,79,80].

5. Conclusions

In conclusion, 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 their interconnection is grounded in Bernstein’s Theorem and the Laplace–Stieltjes transform, and 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.