A remark on the Fourier transform of l^p balls

2022/08/16 by Lind, Martin
#42A38 #42B20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2208.07837

Abstract

We re-examine through an example the connection between the curvature of the boundary of a set, and the decay at infinity of the Fourier transform of its characteristic function. Let B_p⊂ℝ² denote the unit ball of ℝ² in the l^p-norm. It is a consequence of a classical result of Hlawka that for each p∈(1,2], there exists C(p)>0 such that |\widehatχ_Bp(ω)|≤ \fracC(p)|ω|^3/2 (ω∈ℝ², |ω| large). The above estimate does not hold for p=1. Thus, one expects that C(p)→∞ as p→1+; we determine the sharp asymptotic behaviour of C(p) as p→1+.

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