Analytic function with finite integral

Question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an analytic function such that $\int_{\mathbb{R}} \lvert f(x)\rvert dx \in \mathbb{R}$. Is it always true that $\lim_{\lvert x \rvert\rightarrow \infty} f(x) = 0$? I know counterexamples to this when the function is smooth, and I was wondering if they also exist when it is analytic.

Answer 1

Counterexample: The function

$$f(x) = \left (\frac{2+ \cos x}{3}\right)^{x^4}$$

is analytic and integrable over $\mathbb R,$ but $f(2\pi n)=1$ for $n\in \mathbb Z.$ Hence $\lim_{|x|\to \infty}f(x)=0$ fails. Intuitively, the reason for the integrability of $f$ is that the exponent $x^4$ smashes $(2+\cos x)/3$ down to nearly $0$ over much of the line as we head out to $\infty.$ (I haven't made clear why you need $x^4$ rather than, say, $x^2.$ You might want to look up "Laplace's method" to see what's behind this.)