Analysis for the lue-Noise Point Sampling using Kernel Density Model

since Φ̂(ω)Φ̂(−θ) = e−σ2(‖ω‖2+‖θ‖2)/4 = Φ̂(ω−θ)e−σ2(ω·θ∗+ω∗·θ)/4. (9) Assuming X is near equilibrium then, at reasonably low temperature, the power spectrum of the point configuration shows the bluenoise properties, i.e., has a delta at the origin and vanishes in an annulus around it. The approximation function Â(ω) equals to this function times Φ̂(ω) which is zero outside the angulus and hence Â(ω) ≈ δ(ω)Â(0). Therefore, the term in (8) is approximately ‖ω‖2 T e ‖ω‖/2 ( Â(ω)− δρ(ω) )