2020/05/04 by Stefan Steinerberger, Steinerberger, Stefan · 2 citations
Decision Sciences · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2005.01665
openalex publication_date 2020/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f ∈ L2(ℝn) and suppose we are interested in computing its average at a fixed scale. This is easy: we pick the density u of a probability distribution with mean 0 and some moment at the desired scale and compute the convolution u * f. Is there a particularly natural choice for u? This question is studied in scale space theory and the Gaussian is a popular answer. We were interested whether a canonical choice for u can arise from a new axiom: having fixed a scale, the average should oscillate as little as possible, i.e. u = argmin_u supf ∈ L2(ℝn) \frac‖ ∇ (u *f) ‖L2(ℝn)‖f‖L2(ℝn). This optimal function turns out to be a minimizer of an uncertainty principle: for α> 0 and β> n/2, there exists cα, β,n > 0 such that for all u ∈ L1(ℝn) ‖ |ξ|β ⋅ \widehatu‖αL∞(ℝn) ⋅ ‖ |x|α ⋅ u ‖βL1(ℝn) ≥ cα, β,n ‖u‖L1(ℝn)α+ β. For β= 1, any nonnegative extremizer of the inequality serves as the best averaging function in the sense above, β≠ 1 corresponds to other derivatives. For (n, β)=(1,1) we use the Shannon-Whittaker formula to prove that the characteristic function u(x) = χ[-1/2,1/2] is a local minimizer among functions defined on [-1/2,1/2] for α∈ \2,3,4,5,6\. We provide a sufficient condition for general α in terms of a sign pattern for the hypergeometric function 1F2.