(Joint work with Tobias Kaiser; this is essentially a reposting of this post)
The goal of this post is to describe the Hardy field $\H = \Hanexp$ of the expansion $\Ranexp$ of the real field by all restricted analytic functions and the exponential function, based on van den Dries, Macintyre and Marker’s papers on $\Ranexp$ and on LE-series.
In particular, the first paper shows that every function definable in $\Ranexp$ is piecewise given by $\Lanexplog$-terms, and the second paper suggests that the one-variable functions defined by these terms can be described inductively, starting with the set $\E$ of all one-variable functions defined by $\Lanexp$-terms.
For $$\E = \bigcup_{n \in \NN} \E_n,$$ this is done by induction on the “level $n$ of nestedness” of $\exp$ in such a term, distinguishing along the way the subsets $$\P\E = \bigcup_{n \in \NN} \P\E_n$$ of pure germs and $$\M = \bigcup_{n \in \NN} \M_n$$ of monomials: if $n=0$, we set $$\M_0 := \set{x^k:\ k \in \ZZ}$$ and $$\E_0 := \set{(\S F)\left(x^{-1}\right):\ F \text{ is a convergent Laurent series over } \RR},$$ where $\S F$ denotes the geRM at $0$ of the function defined by the sum of $F(X)$.
In this case, $\P\E_0$ is the set of all elements of $\E_0$ with zero constant term. To continue the construction, we also set $$\P\M_0:= \M_0 \cap \P\E_0 = \set{x^k:\ k \in \ZZ \text{ is nonzero}}$$ and $$\P\E_0^\infty:= \set{P\left(x\right):\ P \text{ is a polynomial over } \RR,\ P(0)=0}.$$ Note that the following hold for $n=0$:
- $(\ast)_n$
- $\E_n$ is a differential subfield of $\H$, $\M_n$ and $\P\M_n \cup \{1\}$ are multiplicative subgroups of $\H$, and both $\P\E_n \cup \{0\}$ and $\P\E_n^\infty \cup \{0\}$ are $\RR$-vector subspaces of $\H$.
Crucial to this inductive construction is the following notion:
Definition
A germ $\,f \in \H$ is large if $\lim_{t \to +\infty} f(t) = \pm\infty$, and $\,f$ is small if $\lim_{t \to +\infty} f(t) = 0$.
Note that $\,\P\E_0^\infty$ is exactly the set consisting of all $\,f \in \P\E_0$ whose support contains only large monomials.
For the inductive step of the construction, assume that $n>0$ and the sets $\M_{i}$ and $\P\M_i$, $\E_i$ and $\P\E_i$, as well as $\P\E_i^\infty$ have been defined such that $(\ast)_i$ holds, for $i \lt n$. Then we set $$\P\M_n:= \exp \circ \P\E_{n-1}^\infty$$ and $$\M_n:= \set{a \cdot b:\ a \in \M_{n-1}, b \in \P\M_n \cup \{1\}};$$ in particular, we have $\M_{n-1} \subseteq \M_n$.
Next, we let $\E_n$ be the set of all germs of the form $$F(m_1, \dots, m_k,M_1, \dots, M_l),$$ where $F \in \RR\{X_1, \dots, X_k\}[Y_1, \dots, Y_l]$, the monomials $m_1, \dots, m_k \in \M_n$ are small and the monomials $M_1, \dots, M_l \in \M_n$ are large, and where $\RR\{X_1, \dots, X_n\}$ denotes the set of convergent power series over $\RR$ in $n$ indeterminates.
It is not hard to show the following:
Proposition 1
For nonzero $f \in \E_n$, there exist a unique countable ordinal $\alpha$, unique monomials $m_\beta \in \M_n$ and unique nonzero $r_\beta \in \RR$, for $\beta \lt \alpha$, such that $\frac{m_\beta}{m_\gamma}$ is small for $\gamma \lt \beta \lt \alpha$, and such that the sum $\sum_{\beta \lt \alpha} r_\beta m_\beta$ converges absolutely to $f$ on some interval $(a,+\infty)$.
In the situation of this proposition, we call the monomials $m_\beta$ the principal monomials of $f$ and write $$\M(f):= \set{m_\beta:\ \beta \lt \alpha},$$ and we call $$\lm(f):= m_0$$ the leading monomial of $f$.
Finally, we set $$\P\E_n:= \set{f \in \E_n:\ \text{ every } m \in \M(f) \text{ belongs to } \M_n \setminus \M_{n-1}}$$ and $$\P\E_n^\infty:= \set{f \in \P\E_n:\ \text{ every } m \in \M(f) \text{ is large}}.$$
The main observations about this construction are the following:
Theorem
- As $\RR$-vector spaces we have $\E = \oplus_{n=0}^\infty \P\E_n$.
- $\H = \bigcup_{k \in \NN} \E \circ \log_k$, where $\log_k$ denotes the $k$-th compositional iterate of $\log$.
(For part 2, it suffices to show that $\log \circ f \in \E \circ \log$ for $f \in \E$; to see this, write $f = rm(1+\epsilon)$ with $r \in \RR$ nonzero, $m \in \M$ and $\epsilon \in \E$ small, and note that $\log(1+\epsilon) = G(\epsilon)$, where $G(X)$ is the convergent Taylor series of $\log$ at $1$.)
Definition
- For $f \in \E$, we define the exponential height $\eh(f)$ to be the least $n \in \NN$ such that $f \in \E_n$.
- For general $f \in \H$, we choose the least $k \in \NN$ such that $f \in \E \circ \log_k$, and we set $$\eh(f):= \eh\left(f \circ \exp_k\right) – k,$$ where $\exp_k$ denotes the $k$-th compositional iterate of $\exp$.
Remark
The exponential height is not well behaved under composition: take $g(x) = x+\exp(-x)$ and $f(x) = \exp(x)$. Then $$\eh(f \circ g) = \eh(\exp(x) \cdot (G \circ g)) = 1 \ne 2 = \eh(f) + \eh(g),$$ where $G(X)$ is the (convergent) Taylor series of $\exp$ at $0$.
However, it does behave well in special cases:
Proposition 2
Let $f \in \E$ and $g \in \P\E^\infty$ be such that $g(\infty) = +\infty$. Then
- $\eh(f \circ g) = \eh(f) + \eh(g)$;
- if $f \in \P\E$, then $f \circ g \in \P\E$.
Other useful observations arise out of this construction, and they will be mentioned in later posts as needed.