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What do you mean by analytic? Analytic on the real line, or entire. The method you suggest can be used to obtain an entire example. Let $F$ be a Gaussian. Consider a series $$\sum a_nF(z-c_n)$$ with positive $a_n$ and $c_n$. If $a_n\to+\infty$ does not grow fast, but $c_n$ does grow fast enough, the series converges on every compact in the complex plane, and gives you an entire function with the properties you stated.