Recently, I have learnt from my friend, ZJ, the following result
Assume that is absolutely integrable. Then
The result seems reasonable by the following observation, for example, we consider the first identity when . Then the factor
decays faster then the exponent function . This may be true, of course we need to prove mathematically, because the integrand contains the term which turns out to be a good term since . So here is the trick in order to solve such a problem.
A proof of
To prove this, we split the function under the limit sign into two parts as the following
can be estimated as follows
as . The term
can be estimated as the following
as . Thus, it is clear now to see why the first identity holds.
The rest can be estimated similarly. For the motivation of such identities, we prefer the reader to a paper due to Chang-Qing-Yang published in Duke Math. J. in 2000 [here].