How is the double asymptotic expansion defined?
I can't seem to find it anywhere.
Suppose
$$f(x)\sim \sum_{n=0}^\infty a_n\phi_n(x)$$
as described in the Wikipedia article. How is then, for instance, $f(x,y)$ defined asymptotically? What are its general properties? Any similarities with the one-variable case?
I'm trying to generalize the notion of asymptotic expansion to the two variable case.
Does the following mean anything to you (I just made it up):
$$f(x,y) \sim \sum_{m,n} a_m x^m b_n y^n$$
such that the difference
$$f(x,y)-\sum_{m=0}^{M-1}a_mx^m\sum_{n=0}^{N-1}a_ny^n\in \mathcal{O}(x^my^n)$$
as $(x,y)\rightarrow(0,0)$
Does that make sense?
Generalized asymptotic expansions in one variable very well exist, of course. R. Wong's book (Intro) has both info on the subject, as well as valuable caveats concerning their use.
More to the point, now, note that the "regular" def. of an asymptotic expansion extends naturally to functions of one complex variable. (In fact, that's what Wong, Bender & Orszag & other authors do in their books.) Why is this relevant? Because the complex plane is 2-D.
So how does it work there? Well, the basis functions $\{\phi_n\}_{n\ge0}$ are also scalar functions of a complex variable (duh...) and the limit requirement remains the same but using the metric on the complex plane - i.e., the modulus. Please note, here, that the limit requirement is much less trivial than for functions of a real variable. This is so because so many things can go wrong in taking a limit in more than one dimensions which, in turn, is due to the fact that the limit point can be approached in so many more ways - not just 'from left/right.'
In your made-up example, the first thing to check would be to check that the sequence $\{\phi_{n,m}(x,y):=x^m y^n\}_{m,n\ge0}$ is an asymptotic sequence. Although it would seem reasonable to at least demand that, for any fixed $(n,m)$ $$ \phi_{n,m} = o(\phi_{n',m'}) , \quad\mbox{for all}\quad n'<n \ \& \ m'<m , $$ this cannot be true in the usual `one bounded by the other' sense. Clearly, $x^2 \ne o(y)$ as $(x,y)\to0$ - if you approach the origin from the $x-$axis along which $y\equiv0$, for example. (There's more to it than that, as it's not clear how/whether to order different monomials of the same order. But let's glean over that issue.)
There's an obvious way out of this, namely in mimicking multivariable Taylor series (which asymptotic sequences should generalize). We should, then, allow for multiple monomials of the same order within the basis & order monomials not by trying to bound one by the other but, instead, by recognizing $\phi_{n,m} = O(|(x,y)|^{n+m})$. Then, $\phi_{n,m} = o(\phi_{n',m'})$ in the sense that $|(x,y)|^{n+m} = o(|(x,y)|^{n'+m'})$ & the rest follows from there.
Of course, if you're gonna do all that, why not just switch from Cartesian coordinates $(x,y)$ to spherical $(r,\theta)$ (similarly in more dimensions) and expand in $r$, as $(x,y)\to(0,0)$ means $r\to0$. The coefficients of the asymptotic expansion will depend on the angles. This also mimics Taylor series, since that's pretty much how we reduce the theorem on multivariable Taylor series to the 1-D case. I don't think I've never encountered this in 'real (read: working) life,' though.
(Finally, you might want to also google for 'two-parameter (asymptotic) expansion.')