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For a twice continuously differentiable function $S$, we define the densityfunction of its gradient (derivative in one dimension) $s = S^{\prime}$ as arandom variable transformation of a uniformly distributed random variable using$s$ as the transformation function. Given $N$ values of $S$ sampled at equallyspaced locations, we demonstrate using the method of stationary phase that theapproximation error between the integral of the scaled, discrete power spectrumof the wave function$\phi^{D}_{\tau}=\frac{1}{\sqrt{L}}\exp\left(\frac{iS}{\tau}\right)$ and theintegral of the true density function of $s$ over an arbitrarily small intervalis bounded above by $O(1/N)$ as $N \rightarrow \infty$ ($\tau \rightarrow 0$).In addition to its easy implementation and fast computability in $O(N \log N)$that only requires computing the discrete Fourier transform, our framework forobtaining the derivative density does not involve any parameter selection likethe number of histogram bins, width of the histogram bins, width of the kernelparameter, number of mixture components etc. as required by other widelyapplied methods like histograms and Parzen windows.

Error bounds for gradient density estimation computed from a finite sample set using the method of stationary phase