{"paper":{"title":"Error bounds for gradient density estimation computed from a finite sample set using the method of stationary phase","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"stat.CO","authors_text":"Anand Rangarajan, Karthik S. Gurumoorthy","submitted_at":"2014-04-04T03:43:28Z","abstract_excerpt":"For a twice continuously differentiable function $S$, we define the density function of its gradient (derivative in one dimension) $s = S^{\\prime}$ as a random variable transformation of a uniformly distributed random variable using $s$ as the transformation function. Given $N$ values of $S$ sampled at equally spaced locations, we demonstrate using the method of stationary phase that the approximation error between the integral of the scaled, discrete power spectrum of the wave function $\\phi^{D}_{\\tau}=\\frac{1}{\\sqrt{L}}\\exp\\left(\\frac{iS}{\\tau}\\right)$ and the integral of the true density fu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.1147","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}