On the asymptotic behavior of the Fourier transform of the indicator function of a convex set

Suppose C is a compact, convex subset of Rn, having a smooth boundary AC. Let F(r, 0) be the Fourier transform, in polar coordinates (r= (x2 + + x2)12; =(x1/r, . . ., xn/r)) of the indicator function of the set C, where by the indicator function of C, we mean the function whose value on C is 1, and whose value on the complement of C is 0. Then it is known (cf. [1], [2]) that the function 'D(6) = supr rn + 1)/2 F(r, 0) is bounded on Sn1, provided AC is sufficiently smooth, and has everywhere positive Gaussian curvature. If AC has points of zero curvature, this need no longer be true (cf. [4]). The following, however, remains true.

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