In ordinary analysis, given a sufficiently nice $f:\left[0,\infty\right)\rightarrow\mathbb{C}$, if we can compute the Mellin transform: $$\mathscr{M}\left\{ f\right\} \left(s\right)=\int_{0}^{\infty}x^{s-1}f\left(x\right)dx$$ in closed form computing the inverse mellin transform using the Residue theorem gets us formulae (usually asymptotic, but sometimes exact) for the behavior of $f\left(x\right)$ as $x$ decreases to $0$ and/or as $x$ increases to $\infty$. Ex:
$$\int_{0}^{\infty}x^{s-1}\sum_{n=0}^{\infty}e^{-2^{n}x}dx=\frac{\Gamma\left(s\right)}{1-2^{-s}}$$ implies: $$\sum_{n=0}^{\infty}e^{-2^{n}x}=\frac{1}{2}-\frac{\gamma+\ln x}{\ln2}+\frac{1}{\ln2}\sum_{k\in\mathbb{Z}^{\times}}\Gamma\left(\frac{2k\pi i}{\ln2}\right)x^{-\frac{2k\pi i}{\ln2}}-\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}}{2^{n}-1}\frac{x^{n}}{n!}$$ an (emotionally satisfying!) identity which holds for all x>0.
But, now consider the p-adic analogue: $$\mathscr{M}_{p}\left\{ f\right\} \left(s\right)=\int_{\mathbb{Z}_{p}}\left|\mathfrak{y}\right|_{p}^{s-1}f\left(\mathfrak{y}\right)d\mathfrak{y}$$ where $d\mathfrak{y}$ is the haar probability measure on $\mathbb{Z}_{p}$ and $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}$ is sufficiently nice. Is there a similar interpretation for the inverse mellin transform of $\mathscr{M}_{p}\left\{ f\right\} $ in terms of the asymptotics of $f$?
Continuing in this vein, what about a function of the form:
$$F\left(s\right)=\int_{\mathbb{Z}_{p}}\left(f\left(\mathfrak{y}\right)\right)^{s}d\mathfrak{y}$$
Integral transforms of this form pop up in my current work, but, so far, I can't find anything which explains what these transforms mean. To whit:
(1) The classical mellin transform enjoys "Mapping Theorems" which give a direct correspondence between the asymptotics of the original $f$ near $0$ and $\infty$ and the location and degree of the poles of $f$'s mellin transform. Does such a correspondence exist for the p-adic mellin-type integrals I've listed above? If so, what is it, and/or where can I find literature about it?
(2) Naïvely applying the inverse mellin transform to $F$ (as defined above) doesn't appear to work. In the specific cases I've been working with, either the residue-theoretic approach leads to everywhere-divergent fourier series, or the $F$ in question is an entire function—and, in that case, I have no clue as to how to use $F$ to gain conclusions about the behavior and asymptotics of $f$.
Flajolet et. al. have a wonderful series of articles about the classical mellin transform and its numerous applications to asymptotics. I'm wondering if anything like that exists for mellin-type integrals / integral transforms of real or complex valued functions on the p-adic integers.
I've looked everywhere, and I can't find anything useful about this. All the “relevant literature” is barely applicable, because it is either about Tate's thesis, representation theory, or Tate's thesis as a road to representation theory. I have no interest in representation theory, and it has absolutely nothing to do with my current work. I want to learn about the asymptotic analytic aspects of these p-adic mellin transforms. But I can't find anything like that anywhere. It's extremely frustrating. Answers, help, references, or even mere advice would be much appreciated.