SCHWEITZER 1998 Problem 3

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

related: https://artofproblemsolving.com/community/c7h22594