It is well known that improper integrals don't have to satisfy $\lim_{x\to\infty} f(x)=0$ in order for $\int_a^{\infty} f(x)\,dx$ to converge, for instance $f(x)=\sin(x^2)$. It is also possible to construct such non-negative $f(x)$, and even continuous and non-negative, by taking $f(x)$ to be $0$ everywhere except for triangular spikes with appropriate converging areas.
I was wondering whether we can get such $f(x)$ which is smooth or even analytic. It seems clear that it should be possible to modify the triangular spikes example and get such a smooth $f(x)$ by using bump functions. However, what if we wish for $f(x)$ to be analytic? Bump functions no longer come to the rescue.
To summarize: does there exist a non-negative $f(x)$ which is analytic in $[a,\infty)$ and such that $\int_a^{\infty} f(x)\,dx$ converges, but such that $f(x)$ does not converge to $0$ at infinity?
A comment (without proofs as those can be found in literature) that gives some idea about the complexity of this problem: the result is true if $f$ is the restriction of an entire function of order at most $1$ and finite type - and then it is true in a more general setting, namely that if $f$ has order at most one and is of finite type if it has order $1$ and if $\int_0^{\infty}|f(x)|^pdx<\infty$ for some $p >0$, then $|f(x)| \to 0, x \to \infty$.
(here we can reformulate the condition on $f$ as there is $c>0$ st $|f(z)|e^{-c|z|} \le M$ for all $z$ and some $M >0$)
On the other hand, Carleman approximation theorem implies that for any positive continuous function $g(x)>0, x \in \mathbb R$ (or $x \in I$ any interval), there is an entire function $f$ st $0< g(x)/2 < f(x)<g(x)$ for all $x \in \mathbb R$ (or $x \in I$). This clearly shows that the result fails in general even for analytic functions on $(a, \infty)$ that are restrictions of entire functions in the plane
As asked in the comments a few references; first for the case of an entire function of order at most one, the general result is in the Paley-Wiener circle of ideas and can be found for example in Boas Entire Functions Theorem 6.7.1 p 98
For example if $f^2$ is integrable on the line, the result follows directly from Paley-Wiener and Riemann-Lebesgue and of course if $f$ only integrable on the line, the result above implies $f^2$ integrable and more generally $f^p$ integrable for all $p>1$ since $f^p(x) \le f(x)$ for large enough $|x|$
The Carleman approximation theorem states that for any continuous function $\psi(x)$ on $\mathbb R$ (or some interval on the line and even more general sets in the plane) and any error function $E(x)>0$ there, one can find an entire function $F$ st $|F(x)-\psi(x)| < E(x), x \in \mathbb R$; applying this with $E=3g/4, \psi=g/4$ gives the result mentioned here - more generally one can interpolate any $0<h<g$ continuous on the line with such an $F$ using $E=(h+g)/2, \psi=(g-h)/2$
A simple (pretty much writing down some Taylor series to get the result for $\psi=0$ when we can actually say more and get an entire non-vanishing anywhere in the plane $F$ st $0<F(x)<E(x)$ on the line), and a few algebraic and integral manipulations) direct proof of the result and actually generalized to $\mathbb R^n$ is in the paper Uniform Approximations by Entire Functions by Stephen Scheinberg, while a more general approach including results for other sets than intervals on the line can be found in the book Lectures on Complex Approximations by D Gaier ch IV,3 p 149 and on