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I have seen the formal definitions of big $\mathcal{O}$ and little $o$, and do all right working with them. Still, I have some questions that a good reference might help clear up.

  • In what level of generality can $o$ and $\mathcal{O}$ be defined?
  • It seems that $\mathcal{O}(f(x))$ (in the appropriate context) is an equivalence class of functions, satisfying certain axioms. What is a good algebraic structure to hold these equivalence classes?

My initial thought is that one would want functions taking values in ordered, positive semirings. Perhaps one could use general posets as the domain for these functions. Of course, the more assumptions one adds, the more that can be proven. I'm sure there are lots of other ways to approach it. Is there any reference you know of that treats $\mathcal{O}$ and $o$ in a way resembling what I describe?

CTVK
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A somewhat general setting for this is valuation theory. If $K$ is a field, then a valuation ring is a subring $\mathcal{O} \subseteq K$ with $a \in \mathcal{O}$ or $a^{-1} \in \mathcal{O}$ for all $a \in K^{\times}$. In particular $\mathcal{O}$ is a local ring with maximal ideal $\mathcal{o}=\mathcal{O} \setminus {\mathcal{O}}^{\times}$. Then one can define $\mathcal{O}(a):= a \mathcal{O}$ and $\mathcal{o}(a) := a \mathcal{o}$ for all $a \in K$.

A particular case of that is when $K$ is an ordered field, and $\mathcal{O}$ is a convex subring (containing $1$), e.g. the ring of finite elements in $K$, which are elements $a$ with $-n<a<n$ for some $n \in \mathbb{N}$.

A subcase of this last one is if $K$ is what is called a Hausdorff field. Take $\mathcal{C}$ to be the ring of continuous functions $\mathbb{R} \rightarrow \mathbb{R}$ under pointwise sum and product. Take the quotient $\mathcal{G}$ of $\mathcal{C}$ for the relation $f \equiv g$ if $f(t) =g(t)$ for all sufficiently large $t \in \mathbb{R}$. The equivalence classes are called germs at $+\infty$. Then the pointwise operations factor through the quotient map (the sum/product of germs is the germ of the sum/product), and the resulting ring $\mathcal{G}$ is partially ordered by saying that the germ of $f$ is strictly smaller than that of $g$ if $f(t)<g(t)$ for all sufficiently large $t \in \mathbb{R}$.

A Hausdorff field is a subfield of $\mathcal{G}$, and as a consequence of the IVP for continuous functions, such fields are linearly ordered by the ordering described above. Thus the second paragraph gives you an example of $\mathcal{O}$-$\mathcal{o}$ notion which is close to the classical Landau notion.

An example of Hausdorff field is the set $L$ of germs of functions that can be obtained as combinations of $\exp$, $\log$, and algebraic operations. Another one is or the closure of $L$ under solutions of all linear differential equations of order $1$. This contains many of the (germs of) non-oscillating functions that appear in elementary mathematics.

Note that in these examples, given $a,b \in K^{\times}$, one has $a \in \mathcal{O}(b)$ or $b \in \mathcal{O}(a)$, contrary to what is the case for say arbitrary real-valued functions.

nombre
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  • Thank you for the answer! One example that comes to mind is the subring of $\mathcal{G}$, generated by functions that are either eventually constant, or eventually equal to $f(x)=x+\sin(x)$ or $g(x)=x+\cos(x)$. Is this subring a field? If so, why does this not contradict the statement that Hausdorff fields are linearly ordered? If not, how does it fail the field axioms? – CTVK Feb 06 '23 at 00:08
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    @CTVK This ring is not a field, first of all because it is just the ring generated by those functions and the (germ of) $\frac{1}{\operatorname{id}+\sin}$ is not in this ring. Second of all, this ring contains the germ of $\sin - \cos$, which does not have an inverse in $\mathcal{G}$, for it vanishes on a cofinal subset of $\mathbb{R}$. – nombre Feb 06 '23 at 09:10
  • Aha, thanks again! – CTVK Feb 07 '23 at 18:41