An application of the stationary phase method for estimating probability densities of function derivatives

We prove a novel result wherein the density function of the gradients—corresponding to density function of the derivatives in one dimension—of a thrice differentiable function S (obtained via a random variable transformation of a uniformly distributed random variable) defined on a closed, bounded interval \Omega \subset R is accurately approximated by the normalized power spectrum of \phi=exp(iS/\tau) as the free parameter \tau–>0. The result is shown using the well known stationary phase approximation and standard integration techniques and requires proper ordering of limits. Experimental results provide anecdotal visual evidence corroborating the result.

💡 Research Summary

The paper introduces a novel method for estimating the probability density of the derivative (gradient) of a smooth scalar function S defined on a bounded interval Ω ⊂ ℝ. The authors consider a uniformly distributed random variable X on Ω and define Y = s(X) where s = S′. The density P(u) of Y exists under mild regularity conditions (S is three times continuously differentiable and S″≠0 almost everywhere). Traditionally, P(u) is obtained via the change‑of‑variables formula, yielding P(u) = (1/L) ∑_{x_k∈A_u} |S″(x_k)|⁻¹, where A_u = { x | s(x)=u } and L = |Ω|.

Instead of computing this sum directly, the authors construct a “wave‑function” φ(x)=exp(i S(x)/τ) with a small positive parameter τ. They define a scaled Fourier transform

F_τ(u) = (1/√{2π τ} L) ∫_Ω exp(i S(x)/τ) exp(−i u x/τ) dx

and note that F_τ belongs to L²(ℝ) with unit norm, so P_τ(u)=|F_τ(u)|² is a legitimate probability density candidate. The central result (Theorem 4.2) states that for any u₀ outside a null set C,

lim_{α→0} (1/α) lim_{τ→0} ∫_{u₀}^{u₀+α} P_τ(u) du = P(u₀).

Crucially, the order of limits (first integrate over a small interval, then let τ→0) cannot be swapped; doing so yields a non‑existent limit.

The proof hinges on the method of stationary phase. The phase function Φ(x)=S(x)−u x has stationary points precisely where S′(x)=u, i.e., the points in A_u. Applying Olver’s stationary phase approximation for τ→0 gives

F_τ(u) ≈ (1/√L) ∑_{x_k∈A_u} exp