On the Fourier transform of the indicator function of a planar set.

Suppose C is a compact subset of the plane having a piecewise smooth boundary 8C. Let F(r, 0) be the Fourier transform, in polar coordinates, 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. In 1 of this paper, we shall describe some relationships between geometric properties of C, and the asymptotic behavior of F(r, 0) as r -* oo. In 2, we shall give applications of the results of 1 to some questions in the geometry of numbers.

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