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 Ω\subset R is accurately approximated by the normalized power spectrum of ϕ=exp(iS/τ) as the free parameter τ-->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.

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