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?



Using a quadratic kernel instead of a linear kernel in the Laplace transform
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.
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> 


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


