@MISC{Xu_polynomials, author = {Kun Xu and Li-qian Ma and Bo Ren and Rui Wang and Shi-min Hu}, title = {Polynomials}, year = {} }

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Abstract

cos(φ − µ) ≈ 1 − (φ − µ) 2 /2 The circular Gaussian could be approximated by: g c 2(cos(x − µ) − 1) (φ; µ,λ) = exp[ λ 2 ≈ exp[− (φ − µ)2 λ 2] = g(φ; µ,λ) This means a circular Gaussian can be well approximated by a Gaussian. When φ ∈ [µ − π, µ + π] and λ < π/6, the relative error of this approximation is ≤ 1.3%. 2 Product of two 1D Gaussians A 1D Gaussian g is defined by g(x; µ,λ) = exp(−(x − µ) 2 /λ 2) where µ is the center and λ is the width. Assume two 1D Gaussians g1(x; µ1,λ1) and g2(x; µ2,λ2), it is straightforward to prove that their product is still a Gaussian, given by g1 · g2 = b · g3(x; µ3,λ3), where: µ3 = µ1/λ 2 1 + µ2/λ 2 2 1/λ 2 1 + 1/λ 2 √