SMOOTH APPROXIMATION OF DEFINABLE CONTINUOUS

SMOOTH APPROXIMATION OF DEFINABLE CONTINUOUS FUNCTIONS ANDREAS FISCHER Abstract. Let M be an o-minimal expansion of the real exponential field which possesses smooth cell decomposition. We prove that for every definable open set, the definable indefinitely continuously differentiable functions are a dense subset of the definable continuous function with respect to the o-minimal Whitney topology.

1. Introduction In [3] and [6] it was shown that semialgebraic continuous functions can be approximated by Nash functions with respect to the semialgebraic Whitney topology. Similar approximations were studied for o-minimal structures. We assume the reader to be familiar with the basic concepts of o-minimal structures, as they are presented in [1] or [2]. In the sequel, “definable” always means “definable with parameters in M” for a given o-minimal structure M. Let Rn be endowed with the Euclidean topology. For a definable open set U and p ∈ N ∪ {∞}, we denote by C p (U, Rk ) the definable p times continuously differentiable functions from U to Rk , and by C(U, R) the definable continuous functions from U to R. In all o-minimal expansions of the reals, C p (U, R) ⊂ C(U, R) is dense with respect to the o-minimal Whitney topology if p < ∞, cf. [4]. Here we consider o-minimal expansions M of the real exponential field. In [5] Chapter 8 it was shown that for every definable open U ⊂ Rn , C ∞ (U, R) ⊂ C(U, R) is dense with respect to the o-minimal Whitney topology if (a) M is locally polynomially bounded, and (b) M possesses C ∞ cell decomposition. We show by proving the following theorem, that assumption (a) can be omitted. Theorem 1.1. Let M be an o-minimal expansion of the real exponential field which possesses C ∞ cell decomposition. Let U ⊂ Rn be open and let f ∈ C(U, R). Then, for every ε ∈ C(U, (0, ∞)), there is a g ∈ C ∞ (U, R) with |g(u)−f (u)| < ε(u), u ∈ U. 2. Preliminaries A definable function f : A → Rk is called a C ∞ function, if there are an open definable neighborhood U of A and an F ∈ C ∞ (U, Rk ) such that F |A = f . Definition 2.1. A C ∞ cell in R is either a single point or an open interval. Supposing all cells of Rn are known, then a C ∞ cell in Rn+1 is a set of either the form (h)X := {(x, r) : x ∈ X, r = h(x)} where X ⊂ Rn is a C ∞ cell and h ∈ C ∞ (X, R), Date: 30.01.2007. 2000 Mathematics Subject Classification. Primary 03C64; Secondary 26E10. Key words and phrases. o-minimal structures, exponential function, approximation. Research partially supported by the NSERC discovery grant of Dr. Salma Kuhlmann. 1

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ANDREAS FISCHER

or of the form (f, g)X := {(x, r) : x ∈ X, f (x) < r < g(x)} where X ⊂ Rn is a C ∞ cell and f, g ∈ C ∞ (X, R) ∪ {±∞} such that f (x) < g(x), x ∈ X. The functions used to describe a C ∞ cell Z, we call the defining functions of Z. Note that a C ∞ cell decomposition of R is a finite partition of R into C ∞ cells. A finite partition of Rn+1 into C ∞ cells Z1 , ..., Zr is called a C ∞ cell decomposition, if the set of projections π(Zi ), i = 1, ..., r, is a C ∞ cell decomposition of Rn , where π is the projection onto the first n coordinates. From now on we assume that M possesses C ∞ cell decomposition, that is, for any finite collection of definable sets A1 , ..., Ak ⊂ Rn there exists a C ∞ cell decomposition of Rn partitioning each Ai , i = 1, ..., k. So far, all known o-minimal structures have this property. For a set Z, cl(Z) denotes its topological closure, and ∂Z := cl(Z) \ Z its frontier. Lemma 2.2. Let Z ⊂ Rn be a bounded open C ∞ cell and let f ∈ C(cl(Z), [0, ∞)) satisfy f > 0 on Z. Then there is a g ∈ C ∞ (Z, R) such that (1)

0 < g(z) < f (z), z ∈ Z.

Proof. Let fi , gi , i = 1, ..., n be the defining functions of Z. Then ρ : Z → R, (2) ρ(x1 , ..., xn ) :=

n Y

(xi − fi (x1 , ..., xi−1 ))(gi (x1 , ..., xi−1 ) − xi ), (x1 , ..., xn ) ∈ Z,

i=1

is a C ∞ function which is positive on Z and extends to a definable continuous function ρ : cl(Z) → R with ρ(z) = 0, z ∈ ∂Z. We apply a generalized L Ã ojasiewicz inequality, cf. [2] Corollary C14, to ρ, f and cl(Z), and obtain a definable continuous strictly monotone function φ : R → R with φ(0) = 0 such that 0 < φ(ρ(z)) ≤ f (z), z ∈ Z. By C ∞ cell decomposition, φ restricted to (0, δ) is a C ∞ function for some δ ∈ (0, 1). Let ψ : R → R be defined by ψ(t) := φ(δt2 /(1+t2 )). Then g := ψ◦ρ : Z → R is a C ∞ function and satisfies inequality (1), as δ < 1. ¤ 3. Proof of Theorem 1.1 Proof of Theorem 1.1. the function τ : Rn → Rn which is defined by p By applying p 2 τ (x1 , ..., xn ) = (x1 / 1 + x1 , ..., xn / 1 + x2n ) we can reduce our consideration to bounded open sets. By C ∞ cell decomposition there are finitely many disjoint C ∞ cells Z1 , ..., Zs which cover U such that f |Zi is a C ∞ function, i = 1, ..., s. This partition can be refined in such a way, cf. [1] Chapter 4 Proposition 1.13, that ∂Zi is the union of some of the C ∞ cells. Therefore, each C ∞ cell Zi has a definable open neighborhood Ui which is disjoint to all cells Zj , j 6= i and dim(Zj ) ≤ dim(Zi ). Let Z1 , ..., Zr be the cells of dimension less than n. We order these cells, such that dim(Zi+1 ) ≥ dim(Zi ), i = 1, ..., r − 1. We prove the following statement, which implies the conclusion of Theorem 1.1, by induction on r. For all ε˜ ∈ C(U, (0, ∞)) and F ∈ C(U, R) which satisfy the conditions (a) F |Zi is C ∞ smooth, i = 1, ..., r, and (b) F is C ∞ smooth in U \ ∪ri=1 Zi , there is a g ∈ C ∞ (U, R) such that |g(u) − F (u)| < ε˜(u), u ∈ U. The case r = 0 is evident. We assume that the statement holds for r ≥ 0.

SMOOTH APPROXIMATION OF DEFINABLE CONTINUOUS FUNCTIONS

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Let Z := Zr+1 . After some permutation of the coordinates, Z is the graph of a C ∞ function h = (hd+1 , ..., hn ) : X → Rn−d where X ⊂ Rd is some open C ∞ cell. There also exists an open definable neighborhood U 0 of Z and a function e ∈ C ∞ (U 0 , R) with e|Z = f |Z . In addition, U 0 may be chosen that small that U 0 ∩ Zi = ∅, i = 1, ..., r, and U 0 ⊂ X × Rn−d . We take a definable open neighborhood V of Z which is contained in U 0 such that |e(u) − f (u)| < ε˜(u)/2, u ∈ V . Let ∆ : X → R be the definable continuous function given by (3)

∆(x) := min (dist(h(x), ∪i≤r Zi ), dist(h(x), ∂Z), dist(h(x), ∂V )) , x ∈ X.

By Lemma 2.2 there is a ϕ ∈ C ∞ (X, R) such that 0 < ϕ(x) < ∆(x)/n, x ∈ X. For 0 < s ≤ 1, let Ws be given by (4)

Ws := {(x, y) : x ∈ X, |yi − hi (x)| < sϕ(x), i = d + 1, ..., n} ⊂ V.

The Ws contain Z, and they are C ∞ cells with defining C ∞ functions fi , gi of X, i = 1, ..., d, and fi = hi − sϕ, gi = hi + sϕ, i = d + 1, ..., n. We let σ : R → R be the C ∞ function given by σ(t) = exp(1/t) if t > 0, and σ(t) = 0 if t ≤ 0. Let the functions ψ1 , ψ2 : X × Rn−d → R be defined by (5)

ψ1 (x, y) :=

n Y

σ(yi − hi (x) + ϕ(x))σ(hi (x) + ϕ(x) − yi ), respectively

i=d+1

¶¶ ¶ µ Y µ µ 1 1 ψ2 (x, y) := . σ hi (x) − ϕ(x) − yi + σ yi − hi (x) − ϕ(x) 2 2 i=d+1

Finally, we define the function G : U → R by   ψ1 (u)e(u) + ψ2 (u)F (u) , ψ1 (u) + ψ2 (u) (6) G(u) :=  F (u),

if u ∈ V, otherwise.

By construction, G ∈ C(U, R), G coincides with F outside W1 , and G satisfies (7)

|G(u) − F (u)|