A remark on the Fourier transform of $l^p$ balls

Publication date: 16 August 2022

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} s u b s e t m a t h b b R^{2}

denote the unit ball of

m a t h b b R^{2}

in the

l^{p}

-norm. It is a consequence of a classical result of Hlawka that for each

p i n (1, 2]

, there exists

C (p) > 0

such that

|widehat{chi}_{B_p}(omega)|le frac{C(p)}{|omega|^{3/2}}quad(omegainmathbb{R}^2, |omega| ext{ large}).

The above estimate does not hold for

p = 1

. Thus, one expects that

C (p) i g h t a r r o w i n f t y

as

p i g h t a r r o w 1 +

; we determine the sharp asymptotic behaviour of

C (p)

as

p i g h t a r r o w 1 +

.

Mathematics Subject Classification ID

Singular and oscillatory integrals (Calderón-Zygmund, etc.) (42B20) Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type (42A38)

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