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fourier transform on Zp

Source: miklos schweitzer 1998 q3

September 18, 2021
number theorycomplex analysis

Problem Statement

Let p be a prime and f:Zp→Cf: Z_p \to C a complex valued function defined on a cyclic group of order p. Define the Fourier transform of f by the formula: f^(k)=∑l=0p−1f(l)ei2πkl/p(k∈Zp)\hat f (k) = \sum_{l = 0}^{p-1} f (l) e^{i2\pi kl / p}\qquad(k \in Z_p) Show that if the combined number of zeros of f and f^\hat f is at least p, then f is identically zero.
related: https://artofproblemsolving.com/community/c7h22594
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