Question

Suppose $f$ is a bounded continuous function on $[0,\infty)$ such that $\int_0^\infty f(t) \exp(-xt) \: dt \rightarrow 0$ as $x \rightarrow 0^+$. Does it follow that $\int_0^\infty f(t) \exp(-xt^2) \: dt \rightarrow 0$ as $x \rightarrow 0^+$? Is the reverse implication true?

I suspect that the answer is "no" in both cases, so here's my real (although vague) question: is there a notion of regularity for $f$ (along the lines of the notion of almost-periodicity) such that the two limit-assertions imply each other when $f$ is regular?

Answer 1

There exists a Tauberian theorem of the following form. Suppose that $a\in L^1_{loc}(\mathbb{R}_{>0})$ and for any $x>0$ the integral

$$ G(x):=\int_0^\infty e^{-x^2t^2} a(t) dt $$

exists and satisfies

$$ \sup_{x>0}|G(x)|<\infty, $$

and

$$ \lim_{x\searrow 0} G(x) = A\in\mathbb{R}. $$

The function $G(x)$ is called the Gauss-Weierstrass transform of $a$. Suppose also that the integrals

$$ L(x)=\int_0^\infty e^{-xt} a(t) dt $$

exist for any $x>0$. Then

$$ \lim_{x\searrow 0} L(x)= A= \lim_{x\searrow 0} G(x).$$

Let us point out that if $a\in L^1(\mathbb{R}_{>0})$, then the integrals $G(x)$ and $L(x)$ exist for any $x$ and

$$ A=\int_0^\infty a(t) dt. $$

For a proof see Chap.I, Sec 14, Thm. 21 of the book

S. Bochner, K. Chandrasekharan: Fourier Transforms, Ann. Math. Studies, vol. 19, Princeton University Press, 1949.

Addendum It seems that one of Wiener's Tauberian theorem. (see J. Korevaar: Tauberian Theory. A Century of developments, Theorem 5.1, Chap II).