A (perhaps) surprising fact is that an infinite series of rapidly-decaying functions can sum to a much more slowly-decaying function. For example, the Hermite polynomials $H_n(x)$ span the weighted $L^2$ space $L^2(\mathbb{R}, e^{-x^2} d\mu)$. So if we consider an infinite series of the form $$\sum_{n=0}^\infty c_n\, H_n(x)\, e^{-\frac{1}{2}x^2},$$ then even though each term in the series falls off asymptotically very quickly as $\sim e^{-\frac{1}{2} x^2}$, the series can converge to a function $f(x)$ that falls off much more slowly, e.g. as $f(x) \sim x^{-\left( \frac{1}{2} + \epsilon \right)}$ for arbitrarily small $\epsilon$.
Are there any general restrictions on how much the asymptotics of a function series can differ from the asymptotics of the summand functions? For example, I suspect that it's impossible for a series of exponentially decaying functions to sum to a function that only decays logarithmically (i.e. as $\sim 1/\ln x$), but I'm not sure. And I assume that these restrictions are much tighter if the series is required to converge uniformly rather than pointwise?