On quasi-analytic Denjoy-Carleman classes

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Universit`

a degli Studi di Pisa

Dipartimento di Matematica

Corso di Laurea Magistrale in Matematica

Tesi di Laurea Magistrale

On quasi-analytic

Denjoy-Carleman classes

12 maggio 2017

Candidato

Lapo Dini

Prof.ssa Francesca Acquistapace

Relatore

Universit`a di Pisa

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Abstract

This thesis is a compilatory work on quasi-analytic Denjoy-Carleman functions, meaning classes of real functions defined by bounds on the successive derivatives which contain no flat function. After an introduction to the matter, we include result concerning the algebraic properties of the local rings of germs of quasi-analytic functions and with the geometric properties of quasi-analytic manifolds and sets. Particular importance is given to the failure of the Weierstrass division and preparation theorems in quasi-analytic classes and the related open problem of noetherianity of the local rings is touched upon. On the side of quasi-analytic sets, we report results on resolution of singularities, which can sometimes suffice in lieu of the preparation theorem. We also show how to derive model theoretic properties of quasi-analytic functions from resolution of singularities and why said results are relevant in model theory. We close by relaying some of the more recent developments in the field of quasi-analytic classes.

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Introduction

A quasi-analytic class Q is a class of real functions such that if f ∈ Q and all the derivatives of f vanish, then f is constantly zero, in analogy with analytic continuation. A prime of example of quasi-analytic class are the Denjoy-Carleman classes of functions defined by bounds on the derivatives of the type

|Dα_{f (x)| ≤ AB}|α|_α!m |α|

for some sequence (mk)k∈N. With this work we aim to collect the main results on the

theory of quasi-analytic local rings and of quasi-analytic manifolds and sets and also to show some of the state of the art on quasi-analytic functions. We hope that the reader will gain an understanding of the recurring techniques of the subject and to instill them with at least a modicum of interest in the matter.

The history of quasi-analytic functions began at the turn of the twentieth century with Borel, who, in [8, 9], investigated the behaviour of certain series of rational functions of the form

X A_k (z − ak)nk

, whose sum was completely determined, on a segment, by the knowledge of the derivatives at a point, as is the case for analytic functions. In 1912, Hadamard asked in [28] whether the condition of quasi-analyticity for ultra-differentiable functions could be characterized in terms of a bound on the growth of the derivatives. This question was answered affirmatively by Denjoy in [20] and Carleman in [13], between 1921 and 1926. At first quasi-analytic functions were mainly studied in the setting of partial differential equation and harmonic analysis, but these are not areas that will be significantly touched upon in the coming pages.

New interest in quasi-analytic Denjoy-Carleman classes sparked with the study of the properties of local rings of quasi-analytic function with the methods of analytic geometry. In 1976 Childress proved in [18] that the Weierstrass division theorem fails for

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quasi-analytic Carleman local rings and showed that all quasi-analytic Denjoy-Carleman functions for which the Weierstrass division property holds are hyperbolic polynomials. An important consequence is that one cannot translate to quasi-analytic local rings the classical argument for noetherianity used for analytic functions or formal power series. The question of whether quasi-analytic local rings are noetherian is still open. The work on the division and preparation properties continues to these days with results generalizing the work of Childress to more general quasi-analytic classes and to weaker forms of the preparation and division properties.

A breakthrough in the study of the geometric properties of quasi-analytic functions was reached by Bierstone and Milman in [5], where they proved a version of resolution of singularities in characteristic 0 that applies to many classes of functions, including alge-braic, analytic and quasi-analytic. This result, applied in the quasi-analytic case, implies, among other consequences, that quasi-analytic varieties are topologically noetherian and that Lojasiewicz type inequalities hold for quasi-analytic functions.

This result on resolution of singularities found application in model theory in the early 2000s, when Rolin, Speisseger and Wilkie proved in [42] that restricted quasi-analytic functions give an o-minimal structure. This family of o-minimal structures, which exhibit significantly different behaviour from structures obtained from analytic functions, allowed Rolin, Speisseger and Wilkie to prove that not all o-minimal structures on the real line admit analytic cell decomposition and that there is no largest o-minimal expansion of the real line.

Current research on quasi-analytic functions and quasi-analytic local rings revolves heavily around the open question of noetherianity of the local rings, which is believed to have a negative answer.

Chapter 1 of this thesis introduces the definitions of ultra-differentiable function, quasi-analytic class and Denjoy-Carleman class, while proving many of the classical theorems on the structure of classes of ultra-differentiable classes, due to Cartan and Mandelbrojt [14], Roumieu [43] and Komatsu [32]. It concludes with the Denjoy-Carleman theorem on the injectivity of the Taylor map and the theorem of Carleman on the failure of surjectivity of the Taylor map, when the codomain is restricted to those formal series that satisfy the same growth condition.

In Chapter 2, we consider the Weierstrass division and preparation properties for quasi-analytic functions. The starting point is Childress’s result on the failure of the Weierstrass division theorem for quasi-analytic Denjoy-Carleman classes and his study of hyperbolic polynomials. We then extend our viewpoint to recent results of Elkhadiri, Sfouli [24] and Parusi´nski, Rolin [41] which expand Childress’s result to more general

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quasi-analytic classes, and to the article [2] of Acquistapace, Broglia, Bronshtein, Nicoara, Zobin that treats the failure of Weierstrass preparation when coefficients are allowed to belong to some wider quasi-analytic class.

Chapter 3 deals with the crucial result of Bierstone and Milman on resolution of singularities for quasi-analytic sets. We do not provide the result in its full generality as found in [5], but instead go through the simplified proof provided in [6] in the case of quasi-analytic sets, which suffices for most practical aims. In the end of the chapter we list some of the corollaries to resolution of singularities.

In Chapter 4 the matter shifts slightly to more model theoretic arguments. After a brief introduction of the terms to be used, we follow Rolin’s, Speisseger’s and Wilkie’s proof of the o-minimality and model completeness of the real line augmented with restricted quasi-analytic functions, from [42], with insertions from [23]. In closing we show the proof found in [42] of a theorem of Mandelbrojt on writing a C∞ function as a sum of quasi-analytic functions, and its application to prove that there is no maximal o-minimal structure on the real line.

In Chapter 5 we report on recent developments in the study of quasi-analytic functions. We expose preprints from Nowak [38], Belotto da Silva, Biborski and Bierstone [3] and Bierstone and Milman [7] dealing with quantifier elimination in quasi-analytic structures, resolution of quasi-analytic equations and Malgrange division by quasi-analytic functions, respectively.

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Chapter 1

Denjoy-Carleman classes

The first chapter introduces the notation and main definitions used throughout this work. It also includes many classical results which characterize desirable properties of Denjoy-Carleman classes, which will be needed for later results, in terms of the sequence m defining the class.

1.1 Ultra-differentiable functions

Before proceeding with the definition of an ultra-differentiable function, let us fix some quite standard notation that will shorten the notation for multi-indices. When α = (α1, . . . , αn) ∈ Nn is a multi-index, we will use the notation:

|α| = α₁+ · · · + αn, α! = α1! · · · αn!, xα = xα1 1 · · · xαnn, Dα = ∂α1+···+αn ∂xα1 1 · · · ∂x αn n .

Also, when α, β ∈ Nn _{are multi-indices we will denote:}

α ≥ β if αi≥ βi for i = 1, . . . , n and, if α ≥ β, α − β = (α1− β1, . . . , αn− βn), α β = α! β!(α − β)!.

Definition 1.1. Given a non-decreasing sequence of real numbers m = (mk)k∈N such

that m0 = 1, a C∞ function f on an open set U ⊆ Rn is said to be ultra-differentiable of

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class En(m)(U ) if kDα_{f k} U = sup x∈U |Dα_{f (x)| ≤ AB}|α|_α!m |α| (1.1.1)

holds for every multi-index α ∈ Nn _{and for some constants A, B ∈ R, depending on f .}

When the dimension n and the open set U are clear from context, f will also be said to be of class E (m).

Remark. If we choose the constant sequence of all ones 1 = (1)_k∈N, the relative class E_n(1)(U ) is none other than the class of analytic functions On(U ). In this case the bound

on the derivatives of f is:

kDα_{f k}

U ≤ AB |α|_α!

which is a well-known condition for analyticity.

It is also possible, and in fact quite frequent in the literature, to define the class En(M )(U ) associated to a sequence M = (Mk)k∈N as those functions satisfying:

kDα_{f k ≤ AB}|α|_M

|α| (1.1.2)

It is clear that, if we put Mk= k!mk, the relation between the two possible definitions is

that En(M )(U ) = En(m)(U ), as the ratio |α|!_α! is bounded by 1 from below and by 2n|α|

from above.

Depending on the result one aims to prove, one choice for the defining sequence or the other may result in cleaner notation and we will sometimes use the sequence M in the upcoming proofs.

To ensure a good behaviour of the class En(m)(U ) some conditions are needed on

the sequence m. First and foremost we would like En(m)(U ) to be an R − algebra. A

sufficient condition for this is that the sequence m is logarithmically convex.

Definition 1.2. A sequence m = (mk)k∈N is logarithmically convex if the sequence

of logarithms (log mk)k∈N is convex. Equivalently m is logarithmically convex if the

sequence (mk+1/mk)k∈N is non-decreasing or if the inequality mk+1mk−1≥ m2k holds for

all k ∈ N.

The two consequences of logarithmic convexity which will be used the most are the following:

Lemma 1.3. If m is a logarithmically convex sequence, then mjmk≥ m0mj+k for all

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1.1. ULTRA-DIFFERENTIABLE FUNCTIONS 11

Proof. Let us assume, without loss of generality, that j ≤ k. If j = 0 there is nothing to prove, so we can also assume j ≥ 1. This means that mj/mj−1≤ mk+1mkby logarithmic

convexity, which gives

mjmk≤ mj−1 mj mjmk mk+1 mk = mj−1mk+1

We can now proceed by induction on j to obtain that mjmk≥ m0mj+k holds.

Lemma 1.4. If m = (mk)k∈N is a logarithmically convex sequence and p, q, r ∈ N with

p ≥ q ≥ r, we have the inequality

mr−p_q ≤ mq−p_r mr−q_p .

In particular, when p = 0 and m0= 1 this can be rewritten as

m1/q_q ≤ m1/r_r .

Proof. The proof is an easy induction over r − q and q − p. Applying r − q times lemma 1.3 to mrmr−qp we get

mrmr−qp ≥ mqmr−qp+1.

If we repeat this process q − p times we then have

mq−p_r mr−q_p ≥ mq−p_q mr−q_q = mr−p_q as desired.

Proposition 1.5. If the sequence m is logarithmically convex, the class En(m)(U ) is a

ring.

The proof of proposition 1.5 is quite straightforward and not very interesting, but we have elected to include it nonetheless, in the interest of completion.

Proof. Let f , g ∈ En(m)(U ); we want to show that f + g, f · g belong to En(m)(U ). Note

that associativity, commutativity, and distributivity are obviously inherited from R, since operation in En(m)(U ) are defined pointwise. Closure under addition actually requires

no assumption on the sequence m. We have in fact

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and it easy to estimate this sum. A possible estimation is AfB |α| f α!m|α|+ AgB |α| g α!m|α| ≤ 2 max{Af + Ag}(max{Bf + Bg})|α|α!m|α|.

To show that En(m)(U ) is closed under multiplication, the logarithmic convexity will

be used. First we have:

kDα(f · g)k_U = X β≤α α β Dβf · Dα−βg U ≤ X β≤α α β D β_f U· D α−β_g U

thanks to the generalized Leibnitz rule. Using the bounds for the norms of the derivatives of f and g we obtain: X β≤α α β D β_f U· D α−β_g U ≤ X β≤α α β AfB |β| f β!m|β|AgB |α−β| g (α − β)!m|α−β| ≤ ≤X β≤α α!(AfAg)(max{Bf, Bg})|α|m|β|m|α−β|.

Now we know from lemma 1.3 that the product m|β|m|α−β| is smaller than m0m|α|=

m|α|. Putting it all together we obtain

kDα_{(f · g)k} U ≤

X

β≤α

α!(AfAg)(max{Bf, Bg})|α|m|α| (1.1.3)

and the sum in 1.1.3 has (α1+ 1) · · · (αn+ 1) ≤ 2|α| terms, which are constant with

respect to the summation index. So we can conclude that

kDα(f · g)k_U ≤ (A_fAg)(2 max{Bf, Bg})|α|α!m|α|.

Remark. To prove the previous proposition we do not actually need for the sequence m to be logarithmically convex: it is sufficient for the sequence M to be logarithmically convex, which is a weaker assumption. The proof remains mostly the same, the main difference is in the use of the identityP

β≤α α β = 2

|α| _{for multi-indices.}

The ring structure on the classes En(m)(U ) behaves well with respect to restriction

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1.1. ULTRA-DIFFERENTIABLE FUNCTIONS 13

En(m)(V ) is a ring homomorphism.

The definitions that we have given for now cover only the case of ultra-differentiable functions on some open set U ⊆ Rn. Since we will be interested in functions defined on an arbitrarily small neighbourhood of some point x ∈ Rn, we will mostly use stalks of germs of functions at a point.

Definition 1.6. Fix a point x0 in the space Rn and order the set of all neighbourhoods

of x0 by inclusion, so that it is a directed set. The stalk of germs of ultra-differentiable

functions of class E (m) at the point x0, denoted by En,x0(m) is the direct limit

E_n,x₀(m) = lim_−→E_n(m)(U ) where U varies over all open neighbourhoods of the point x0.

The elements of En,x0(m) are not functions but rather germs of functions. Given

a function f defined in a neighbourhood U of the point x0, we remind that the germ

f ∈ En,x0(m) is the equivalence class [(f, U )]∼ of the couple (f, U ) by the equivalence

relation

(g1, V1) ∼ (g2, V2) if there exists W ⊆ (V1∩ V2) such that g1|W ≡ g2|W.

We remember that for a germ of function f at the point x0 the value f (x0) is well defined

and equal to the value f (x0) for any representative f of f . By the properties of the

direct limit, the stalk of germs En,x0(m) inherits the R-algebra structure from the classes

En(m)(U ). Throughout the rest of this text, we will, with an abuse of notation, denote

both a function f and its germ at a point f with the letter f without bold face. We will also write En(m) for En,x0(m) without specifying the point x0, which will usually be

assumed to be the origin of Rn.

Given a class En(m), we can consider the map which maps each germ about 0 to its

Taylor series at 0 ˆ· : En(m) → C[[x1, . . . , xn]] (1.1.4) f (x) 7→ ˆf (x) = X α∈Nn Dα_{f (0)} α! x α _(1.1.5)

We can define sub-rings Fn(m) of the ring of formal series Fn = C[[x1, . . . , xn]] by

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m = (mk)k∈N

X

α∈Nn

cαxα∈ Fn(m) if there are A, B > 0 such that |cα| ≤ AB|α|m|α| for all α ∈ Nn.

It is clear that the Taylor series map ˆ· sends En(m) in Fn(m). It is a well known result

that ˆ· is injective when En(m) = On, an important question is in which other cases is it

injective.

Definition 1.7. A class En(m) such that En(m) ) Onand ˆ·: En(m) → Fn(m) is injective

will be called a quasi-analytic class.

Since an eventual failure of injectivity of ˆ· corresponds to the existence of a germ f such that Dαf (0) = 0 for all α ∈ Nn but f 6≡ 0, a quasi-analytic class is a class En(m)

which contains no flat germs. The Denjoy-Carleman theorem will characterize such classes, but before reaching that we will go on studying other properties of ultra-differentiable classes En(m).

1.2 Inclusions between classes of ultra-differentiable

func-tions

Given two logarithmically convex sequences m, m0, we ask ourselves when there can be an inclusion En(m) ⊆ En(m0). If there exists a positive constant C ∈ R such that the

condition mk ≤ Ckm0k holds for all k ∈ N, for f ∈ En(m) we have

kDαf k_U ≤ AB|α|α!m|α| ≤ A(B · C)|α|α!m0|α|,

from which we obtain f ∈ En(m0) and En(m) ⊆ En(m0). The requirement mk≤ Ckm0_k

for all k ∈ N can be expressed equivalently as

sup

k∈N

(mk/m0k)1/k< ∞.

It was proved by Cartan and Mandelbrojt in [14] that this condition is not only sufficient, but also necessary. This is easy to see in the case of the rings of formal series Fn(m) and

Fn(m0). Assume that Fn(m) ⊆ Fn(m0) and consider the formal series

F (x) = X

α∈Nn

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1.2. INCLUSIONS BETWEEN CLASSES OF ULTRA-DIFFERENTIABLE FUNCTIONS15

Clearly F ∈ Fn(m), which, by our assumption, implies F ∈ Fn(m0). Then we must have

m|α| ≤ AC|α|m0|α|

by the definition of Fn(m0) and we have our claim. In the case of ultra-differentiable

functions, Cartan’s and Mandelbrojt’s idea is how to find a function f ∈ En(m) whose

derivatives are big enough to allow us to claim mk≤ Ckm0k. Unfortunately there is no

guarantee that there is a function f ∈ En(m) whose Taylor series is exactly F , as we will

see later in this chapter, so the real difficulty lies in finding a suitable function f . To define the function f we are looking for, it will be helpful to rephrase the condition mk≤ Ckm0k as a condition on certain functions of a real variable r. For this purpose,

it will be more practical to use the sequence M = (Mk)k∈N= (k!mk)k∈N instead of the

sequence m and a similar sequence M0 instead of m0. Let the function S of real variable r ≥ 0 be defined by: S(r) = sup k∈N rk Mk ; (1.2.1)

the analogous function for M0 will be denoted S0(r).

Consider the sequence µ = (µk)k∈N defined by µ0 = 1, µk = Mk/Mk−1 for k ≥ 1;

by the logarithmic convexity of the sequence m, we have that µ is increasing and that µk ≥ k for all k ∈ N. If we fix r ≥ 0 real, we have S(r) = rk0/Mk0 where k0 is the

smallest integer such that µk0+1 ≥ r. Note that such k0 is always strictly smaller than r,

as µk≥ n implies µdre ≥ rm so that we have

S(r) = sup k∈N rk Mk = max k≤r rk Mk .

Thanks to this remark we can write the function S piecewise as

S(r) =                      1 M0 if r ≤ µ1 r M1 if µ1 < r ≤ µ2 .. . rk Mk if µk< r ≤ µk+1 .. . (1.2.2)

Once we have expressed the function S this way it is easy to see that S is non-decreasing and continuous, as rk−1/Mk−1 = rk/Mk when r = µk.

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We also have a way to recover the sequence M from the function S, by Mk = sup r≥0 rk S(r) = supr≥k rk S(r).

To see this, write rk/S(r) as rk−k0_M

k0 where k0 is as above. From 1.2.2 we see that the

function rk/S(r) is increasing for k > k0, constant for k = k0 and decreasing for k < k0

and that these conditions are equivalent, respectively, to r ≤ µk, µk < r ≤ µk+1, and

r > µk+1. Since µk ≥ n, the supremum, which is, in fact, a maximum, is reached for

µk≤ r ≤ µk+1 and it equals exactly Mk.

We can now express the condition Mk≤ CkMk0 through the respective functions S(r)

and S0(r).

Proposition 1.8. The two conditions 1. Mk≤ Mk0 for all k ∈ N

2. S(r) ≥ S0(r) for all r ≥ 1 are equivalent.

Proof. Assume (1). For a fixed r ≥ 1 take k0 ∈ N such that µ0k0 ≤ r < µ

0 k0+1. Then S0(r) = r k0 M_k0 0 ≤ r k0 Mk0 ≤ max k≤r rk Mk = S(r).

Now assume (2). For a fixed k ∈ N take r0 ∈ R between µk and µk+1. Then

Mk= rk₀ S(r0) ≤ r k 0 S0_(r 0) ≤ sup r≥k rk S0_(r) = M 0 k. If we let S_C0 (r) = max k≤rr k_/Ck_M0 k, the condition Mk≤ CkMk0 for some C ∈ R, C > 0 is equivalent to S(r) ≥ S_C0 (r) for some C ∈ R, C > 0.

It is enough for the inequality S(n) ≥ S0_C(n) to hold for every positive integer n. To pass to the general case from the integer case, fix r ∈ R and let n ∈ N with n < r < n + 1.

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We have

S(r) ≥ S(n) ≥ S_C0 (n) ≥ S0_2C(r), which is the result we aimed for, with constant 2C.

We now go on to obtain S(n) ≥ S_C0 (n) for integer n.

Lemma 1.9. If S(r) and S0(r) are as above and ν is an increasing sequence of integers, one can find a subsequence (νi)i∈N and a real number D > 0 such that

S(νi) ≥ SD0 (νi)

holds for every index i.

Before proceeding with the proof, we will show how to pass from the lemma to the inequality for all integers. First, note that if an inequality of this kind holds on all but finitely many integers, then, up to choosing a suitably larger constant C, it holds on all integers.

So, let ν(0) be an increasing sequence of integers on which S < SD00 for a fixed D0 > 0.

By the lemma, we can find a subsequence ˜ν(0) and a constant D1, greater than D0, such

that the inequality holds on ˜ν₍₀₎ with constant D1. If the set of integers in ν(0) such that

S < S_D0

1 is finite, we can conclude. Otherwise we can find an increasing sequence ν(1) in

ν(0) where S < SD0 1.

If this process does not stop by a certain ¯k, we can find by induction a family of subsequences {ν(k)} indexed by k ∈ N and an increasing sequence (Dk)k∈N which we can

assume is unbounded. Now we define the diagonal sequence ν = (νk)k∈N= (ν(k),k)k∈N.

Since the sequence (Dk)k∈N is unbounded the lemma cannot hold for ν and thus we have

a contradiction. So there must be a constant C > 0 such that S ≥ S_C0 on all integers. Proof. In showing the proof of this lemma we will closely follow Cartan and Mandelbrojt [14].

Let ν be an increasing sequence of integers, we need to find a subsequence (νi)i∈N of

ν and a number D > 0 such that S(νi) ≥ S_D0 (νi). We define the subsequence (νi)i∈N of ν

by recursion: ν0 is the smallest integer of ν and νi+1 is the smallest integer of ν such

that νi+1> νi and

νi+1> max Mνi+1 Mp Mνi+1 M0 p for all p ≤ νi (1.2.3) holds.

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From the definition of the function S, evaluated for r = νi+1 we have S(νi+1) = max k≤νi+1 ν_i+1k Mk .

If we limit ourselves to the case k = ν1+ 1 we have

[S(νi+1) ≥

ννi+1

i+1

Mνi+1

from which we obtain

ννi

i+1

S(νi+1)

≤ min(M_p, M_p0) for all p ≤ νi,

thanks to our choice of νi+1 in 1.2.3.

Let now j ≥ 0 be a fixed index; if p < nj we have: ∞ X i=j+1 np_i S(ni) = ∞ X i=j+1 nni−1 i S(ni) 1 nni−1−p i ≤   ∞ X i=j+1 1 nni−1−p i  · min(Mp, Mp0) ≤ 2 min(Mp, Mp0) (1.2.4) We will now proceed to the construction of a function f (x) of class En(m) ⊆ En(m0)

which will lead us to our conclusion. Let Tn(x) denote the n-th Chebyshev polynomial

on the interval I = (−1, 1),

Tn(x) = cos(n arccos(x)),

and put for every index i ≥ 1

Zi(x) = 1 2(Tni−1(x) + Tni(x)) . We now define f (x) as f (x) = ∞ X i=1 Zi(x) S(ni) .

By the known properties of Chebyshev polynomials, the inequalities np_i

e p

≤ |Z_i(p)(0)| ≤ np_iand (1.2.5) |Z_i(p)(x)| ≤ Kpnp_i (1.2.6)

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hold for p ≤ ni, for x in some closed subinterval I0 ⊆ I and for some K depending on

the interval I0. From this inequalities we have that the series defining f (x) converges uniformly on I0, along with all of its derivatives. More precisely, if j is the smallest integer such that nj ≥ p one has:

|f(p)_{(x)| ≤ K}p   np_j S(nj) + ∞ X i=j+1 np_i S(ni)  .

From this, since np_j/S(nj) ≤ Mp by the definition of S(r), and using inequality 1.2.4 one

has:

|f(p)(x)| ≤ 3KpMp

and as such the function f (x) is of class En(m).

Since we assumed the inclusion En(m) ⊆ En(m0), the function f (x) must also be of

class En(m0), and there exists a B > 0 such that

|f(p)(0)| ≤ BpM_p0.

From here our conclusion is in reach. Assume that nj−1< p ≤ nj, from the inequalities

1.2.5 we obtain |f(p)(0)| ≥ 1 ep np_j S(nj) − ∞ X i=j+1 np_i S(ni) .

Since f ∈ En(m) we also must have

np_j S(nj)

≤ ep(Bp+ 2)M_p0 ≤ CpM_p0

for some C > 0. This does it in the case nj−1< p ≤ nj; if p ≤ nj−1 the inequalities still

hold because of equation 1.2.4 Putting all together we than have

SC(nj) = max p≤nj np_j Cp_M0 p ≤ S(nj)

which is what we wanted to prove.

Putting together all the results of this section we have the proof of the following

Theorem 1.10. Given two logarithmically convex sequences m, m0, En(m) ⊆ En(m0) if

and only if sup

k∈N

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From lemma 1.4 we know that the sequencem1/k_k

k∈N is non-decreasing, as such our

classes En(m) always contain the analytic functions. In view of the preceding theorem

we can also formulate the following:

Corollary 1.11. The class En(m) strictly contains the analytic class On if and only if

lim

k→∞m 1/k k = ∞.

1.3 Composition, inverses and derivation

To ensure that we can work meaningfully on En(m) we will also need the class to have a

good behaviour with respect to some common operations on C∞ functions. Explicitly, we will require the following conditions to be satisfied:

1. En(m) is closed under composition;

2. En(m) is closed under implicit and inverse function;

3. En(m) is closed under derivation and division by a coordinate;

4. En(m) is quasi-analytic.

We will call a class En(m) that satisfies the above requirements a Denjoy-Carleman class.

Since the first three properties also make sense for the ring Fn(m), and we would like

F_n(m) to fulfill them too.

In this sections of this first chapter we will show that if the sequence m is logarithmi-cally convex and fulfills the conditions

sup k∈N mk+1 mk 1_k < ∞,

the class En(m) and the ring Fn(m) have properties (1)-(3). The problem of

quasi-analyticity will instead be faced in the last section of this first chapter.

Proceeding in the order in which we listed our desirable properties, we turn now to show the proof closure under composition. More precisely we mean that for any f = (f1, . . . , fp) ∈ (En(m))p such that fi(0) = 0 for all i = 1, . . . , p and g ∈ Ep(m), the

composite function g(f1, . . . , fp) = g ◦ f belongs to En(m). This result is due to Roumieu

[43] and we will follow his proof.

Theorem 1.12. If m is a non-decreasing logarithmically convex sequence, f = (f1, . . . , fp)

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1.3. COMPOSITION, INVERSES AND DERIVATION 21

Proof. Fix f = (f1, . . . , fp) ∈ (En(m))p, with fi(0) = 0 for all i = 1, . . . , p, and fix

g ∈ Ep(m). Also fix open sets W ⊆ Rn and V ⊆ Rp such that f is defined on W , 0 ∈ W ,

g is defined on V , 0 ∈ V and V ⊆ f (W ). Put U = f−1(V ). Assume that

kDαfikU ≤ α!m|α| and kDαgkV ≤ α!m|α|; (1.3.1)

we need to find A, B > 0 such that

kDα(g ◦ f )k_U ≤ AB|α|α!m|α|.

Note that we can assume the coefficients to be 1 in 1.3.1 in view of theorem 1.3.

To do this we will obtain a bound for the Taylor series of (g ◦ f ) from the known bounds for the Taylor series of f and g.

If we develop the Taylor series of f and g at 0, using the notation X = x1+ · · · + xn

and Y = y1+ · · · + yp we obtain

ˆ

fi(x) ≤ m1X + m2X2+ · · · + mkXk+ . . .

ˆ

g(y) ≤ m1Y + m2Y2+ · · · + mkYk+ . . . .

If we denote F (x) = ˆf1(x) + · · · + ˆfp(x) and sum over i the inequalities for the series ˆfi

we also have F (x) ≤ p m1X + m2X2+ · · · + mkXk+ . . . . Combining the development for g with the one for F , we obtain

ˆ

g(f (x)) ≤ m1p(m1X + m2X2+ · · · + mkXk+ . . . ) +

+ m2p2(m1X + m2X2+ · · · + mkXk+ . . . )2+ · · · +

+ mhph(m1X + m2X2+ · · · + mkXk+ . . . )h+ . . . .

The formula for the h-th power the power series (1 + Z + Z2+ · · · + Zk+ . . . ) is (1 + Z + Z2+ · · · + Zk+ . . . )h = ∞ X j=0 j + h − 1 j Zj, multiplying both terms by Zh we get

(Z + Z2+ · · · + Zk+ . . . )h = ∞ X j=0 j + h − 1 j Zj+h.

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Applying this and the inequality

mjmk≤ m1mj+k−1,

which we have from lemma 1.3 to

(m1X + m2X2+ · · · + mkXk+ . . . )h we obtain (m1X + m2X2+ · · · + mkXk+ . . . )h≤ mh−11 ∞ X j=0 j + h − 1 j mj+1Xj+h,

where the inequality holds term by term. To see this, note that the (j + h)-th term in the development of the series will have as a coefficient the product mk1· · · mkh of h elements

of the sequence m and that mk1 + · · · + mkh must equal j. Now, by repeatedly applying

the lemma we get mk1· · · mkh ≤ m

h−1 1 mj+1.

From this we can get a bound for the coefficient ck of Xk in the development of g ◦ f .

We have ck≤ k X j=1 pjk − 1 k − j mj−1₁ mjmk−j+1

and applying again lemma 1.3 we get

ck≤ k X j=1 pjk − 1 k − j mj₁mk.

Since both p, m1≥ 1, we can bound pj and mj1 with pk and mk1 respectively, to arrive to

ck≤ (p · m1)kmk k X j=1 k − 1 k − j ≤ (2 · p · m1)kmk

We finally have for the derivative of g ◦ f

kDα(g ◦ f )k_U ≤ (2 · p · m1)|α|α!m|α|

and the theorem is proved.

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functions f and g, the same proof can be applied to the case of Fn(m).

From theorem 1.12 we can obtain multiplicative inverse of a non-vanishing function f ∈ En(m) also belongs to En(m).

Corollary 1.13. If m is a non-decreasing logarithmically convex sequence, f ∈ En(m)

and f (0) 6= 0, then _f1 ∈ E_n(m).

Proof. The function 1_x is analytic wherever it is defined, in particular it belongs to the class E1(m). Then the composition _f1 also belongs to En(m) when it is defined.

Thanks to this corollary we can also deduce some information on the algebraic properties of the ring En(m).

Corollary 1.14. If m is a non-decreasing logarithmically convex sequence, En(m) is a

local ring with maximal ideal M = {f ∈ En(m)|f (0) = 0}.

Proof. M is clearly an ideal of En(m). If f ∈ En(m) \ M we have f (0) 6= 0; by corollary

1.13 we have that 1_f ∈ E_n(m) so that f is a unit in En(m). By a well known property of

local rings this means that En(m) is local with maximal ideal M.

As we proceed through the list of properties that we listed at the beginning of this section; we now turn our attention to closure under inverse and implicit function of the class En(m). By this we mean that when a function f ∈ En(m) is under the hypotheses of

the classical Inverse function theorem or Implicit function theorem, then also the inverse mapping f−1 or the implicit function g belong to the class E1(m). This result is due to

Komatsu [32] and we will follow that article in our proof.

Theorem 1.15 (Inverse function). If m is a non-decreasing logarithmically convex sequence, the function

f = (f1(x1, . . . , xn), . . . , fn(x1, . . . , xn)) belongs to (En(m))n with f (0) = 0, and its

Jacobian determinant ∂(f1, . . . , fn) ∂(x1, . . . , xn) = det ∂fi ∂xj

does not vanish in the origin, then there exist open neighbourhoods U0 and V0 such that f

is defined on U0, is a diffeomorphism from U0 onto V0 and its inverse is of class En(m).

Proof. As we already said, the proof is taken from [32]. By the Inverse mapping theorem for C∞ mappings, we can find open sets U0 and V0 such that F : U0 → V0 is a C∞

-diffeomorphism. We need to show that the map F−1 = (g1, . . . , gn) is of class En(m)

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Let (aij) be the inverse matrix of (∂fi(0)/∂xj); we define functions ϕi(x) = xi+ n X j=1 aijfj(x)

for each i = 1, . . . , n. each function ϕi is a linear combination of functions of class En(m)

and as such is of class En(m) itself. This means that we can write

kDαϕik_U₀ ≤ AB|α|α!m|α|

for each i = 1, . . . , n. As a consequence, if we denote by X = x1+ · · · + xn as in the

proof of theorem 1.12, the Taylor series ˆϕi of ϕi fulfills

ˆ ϕi(x) ≥ A ∞ X k=2 mkBkXk (1.3.2) for each i = 1, . . . , n.

The components gi of F−1 can be obtained as the solutions of the system of equations

gi(y) = n

X

j=1

aijyj + ϕi(g1(y), . . . , gn(y)).

Now let C an uniform bound for |aij| and Y = y1+ · · · + yn, passing to the Taylor series

and using the inequality 1.3.2 we get

ˆ gi(y) ≤ CY + A ∞ X k=2 mkBk(ˆg1(y) + . . . ˆgn(y))k.

We now define a formal series ψ(Y ) as the solution of the equation

ψ(Y ) = CY + A

∞

X

k=2

mkBknkψ(Y )k,

so that each gi(y) is majorized by ψ(Y ).

Now we will apply the Lagrange inversion theorem to the series ψ(Y ) so that we can write

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where the coefficients bh are given by the formula

bh = 1 h!    d dt h−1 t t − AP k≥2 mkBknktk !h    t=0 . (1.3.3)

Developing the rational function t/(t − AP

k≥2

mkBknktk) in the above expression

through its formal series we have

t t − A P k≥2 mkBknktk !h =   ∞ X j=0 nAB ∞ X k=1 mk+1Bknktk !j  h

We now consider to cases separately; for k < h we have the inequality mk+1 ≤

Γk(mh)k/(h−1) with Γ = m(h−k−1)/k(h−1)1 thanks to lemma 1.4, which gives us

  ∞ X j=0 nAB h−1 X k=1 mk+1Bknktk !j  h ≤   ∞ X j=0 nAB h−1 X k=1 Γm1/(h−1)_h Bnt k !j  h .

In the case k ≥ h instead, we remark that the variable t in equation 1.3.3 is raised to a power of at least k and will be derived only h − 1 times,such terms will then vanish when we evaluate the derivative for t = 0. As such we will proceed in our calculation leaving the infinite sum over k in our formula, knowing that this will not affect the result.

To recap, we have gone from   ∞ X j=0 nAB ∞ X k=1 mk+1Bknktk !j  h to   ∞ X j=0 nAB ∞ X k=1 Γm1/(h−1)_h Bntk !j  h

We now recall the formula

(1 + x + x2+ · · · + xk+ . . . )h = ∞ X k=0 k + h − 1 k xk (1.3.4)

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that we already used in the proof of theorem 1.12. Applying it here, we get   ∞ X j=0 (nAB)j ∞ X k=0 j + k − 1 k Γm1/(h−1)_h Bntj+k   h .

Now we will change the indicization of our series and use i = j + k and j ≤ i instead of j and k   ∞ X i=0 nAB Γm1/(h−1)_h Bnt i_Xi j=0 i − 1 j − 1 (nAB)j−1   h = = ∞ X i=0 nAB(nAB + 1)i−1Γm1/(h−1)_h Bnti !h ≤ ∞ X i=0 (nAB + 1)Γm1/(h−1)_h Bnti !h .

Applying again the formula 1.3.4 for the power of a series to obtain

∞ X i=0 i + h − 1 i (nAB + 1)Γm1/(h−1)_h Bnti. (1.3.5) Since, by 1.3.3, we are interested in the (h − 1)-th derivative of series 1.3.5 evaluated at 0, we only need to know the term in th−1, which is

2h − 1 h − 1 (nAB + 1)Γm1/(h−1)_h Bnth−1. Applying 1.3.3 we get bh ≤ 1 h 2h − 1 h − 1 ((nAB + 1)ΓBn)h−1mh≤ (4(nAB + 1)ΓBn)h−1mh.

If we turn back to considering the functions gi, we have

ˆ gi ≤ ψ(Y ) ≤ ∞ X k=1 mk(4(nAB + 1)ΓBn)k−1CkYk and kDαgikU0 ≤ (4(nAB + 1)ΓBn) |α|−1_C|α|_α!m |α|, and F−1 is of class En(m).

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implicit function theorem, which will only be stated.

Theorem 1.16 (Implicit function). If m is a non-decreasing logarithmically convex sequence, the function

f = (f1(x1, . . . , xp, y1, . . . , yq), . . . , fq(x1, . . . , xp, y1, . . . , yq) belongs to (Ep+q(m))q with

f (0) = 0 and the determinant

∂(f1, . . . , fn)

∂(y1, . . . , yn)

= det ∂fi ∂yj

does not vanish in the origin, then there exist open neighbourhoods U0 and V0 and a

unique function g : U0→ V0 belonging to (Eq(m))p such that f (x, g(x)) ≡ 0 on U0.

As was the case for theorem 1.12, theorem 1.15 mainly involves the Taylor series of f and as such its proof can also be applied to Fn(m).

To close this section we will show that the class En(m) is closed under derivation and

division by a coordinate when m is logarithmically convex and

sup k∈N mk+1 mk 1 k < ∞.

By closure under derivation we mean that if f ∈ En(m), then Dαf ∈ En(m) for

all α ∈ Nn. This is equivalent to asking that the first derivatives _∂x∂f

i are in En(m) for

i = 1, . . . , n, since higher order derivatives are obtained iteratively.

If we denote by m+1 = (m+1_k )_k∈N the shifted sequence m+1_k = mk+1, all the first

derivatives _∂x∂f

i of a function f ∈ En(m) clearly belong the class En(m

+1_{) associated to}

the shifted sequence. We can apply theorem to this case to see when En(m+1) ⊆ E1(m):

Corollary 1.17. Given a non-decreasing, logarithmically convex sequence m, the class E_n(m) is closed under derivation if sup

k∈N _m k+1 mk 1_k < ∞.

Since the result of theorem is true for formal series, this corollary also applies to Fn(m).

By closure under division by a coordinate we mean that if f ∈ En(m) and

f (x1, . . . , xi−1, 0, xi+1, . . . , xn) ≡ 0, there is a function h(x) ∈ En(m) such that the

equality f (x) = xih(x) holds.

This requirement is equivalent to closure under derivation, as we can see from the fundamental theorem of calculus, by writing

f (x1, . . . , xn) − f (x1, . . . , 0, . . . , xn) = xi Z 1 0 ∂f ∂xi (x1, . . . , txi, . . . , xn)dt. (1.3.6)

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From this equation we can also obtain another characterization of the maximal ideal M of E_n(m).

Corollary 1.18. The maximal ideal M ⊆ En(m) is generated by x1, . . . , xn.

Proof. Let f ∈ M ⊆ En(m) so that f (0) = 0. By 1.3.6

f (x1, . . . , xn) = f (x1, . . . , xn−1, 0) + xnfn(x)

for some function fn∈ En(m). Repeating this process for all other variables x1, . . . , xn−1,

we reach

f (x) = x1f1(x) + · · · + xnfn(x) + f (0)

and, since f (0) = 0, f ∈ (x1, . . . , xn).

1.4 The Denjoy-Carleman theorem

In this section we will turn to look at some of the properties of the map

T : En(m) → Fn(m) f (x) 7→ T (f )(x) = X α∈Nn Dαf (0) α! x α

which maps each function f in En(m) to its Taylor series T (f ) in Fn(m). We will usually

denote the Taylor series of a function f by ˆf , instead of writing T (f ).

For starters, we will show a proof of the Denjoy-Carleman theorem, which gives necessary and sufficient conditions for the injectivity of the map ˆ·. We will then also show a proof of another theorem of Carleman that claims that the map ˆ· cannot be both injective and surjective (onto Fn(m)), with the exception of the analytic class.

The condition on the sequence m that ensures quasi-analyticity, and is equivalent to it, is X k∈N mk (k + 1)mk+1 = ∞,

which gives a bound on the growth of the sequence m. It will be enough to show the proof of the Denjoy-Carleman theorem for the case of dimension 1.

To see that this is equivalent to the general case, note that, if f ∈ En(m) is a non-zero

function whose derivatives of any order vanish at 0, we can find a line L through the origin such that f does not vanish identically on L. Now f|L is a non-zero function

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1.4. THE DENJOY-CARLEMAN THEOREM 29

of a single variable which is flat at 0. Conversely if f ∈ E1(m) is non-zero and all its

derivatives vanish at 0, the function of n-variables f (x2₁+ · · · + x2_n) is not trivial, flat at the origin and belongs to En(m) by theorem 1.12.

In the proof of this theorem it will be expedient to use the sequence M = (Mk)k∈N =

(k!mk)k∈N in addition to the sequence m. We will also need the function S(r) =

sup

k≤r

(rk/Mk) already defined in section 2 of this chapter.

Theorem 1.19 (Denjoy-Carleman). If m = (mk)k∈N is a logarithmically convex

non-decreasing sequence with m0 = 1, then the following are equivalent:

i. The class E1(m) is quasi-analytic;

ii. ∞ R 1 log S(r) 1+r2 dr = +∞; iii. ∞ P k=0 mk (k+1)mk+1 = ∞ P k=0 Mk Mk+1 = +∞.

The original proofs of this theorem can be found in [10, 12, 20]. The proof that follows can be found in many analysis textbooks. Our main reference will be [44, pp. 380-383], with [30, pp. 127-130] as an auxiliary source.

Proof. Proof that (ii) implies (i). Assume that E1(m) is not quasi-analytic, we claim that

we can find a non-trivial function f ∈ E1(m) such that f is defined on an open set U and

has compact support in U .

To see this, let g ∈ E1(m) such that g(k)(0) = 0 for all k ∈ N but g(x0) 6= 0 for some

x0> 0, so that E1(m) is not quasi-analytic. The function h(x) defined by h(x) = g(x)

for x > 0 and by h(x) = 0 for x ≤ 0 is still of class E1(m). Define f (x) = h(x)h(2x0− x);

the function f (x) is still of class E1(m) and it has compact support.

Up to an affine rescaling, we can assume that suppf ⊆ [0, H] for some H > 0 and that kf(k)k_[0,A] ≤ 2−kk!mk for all k ∈ N. Define

F (z) = H Z 0 f (t)eitzdt G(w) = F i − iw 1 + w

By the theory of Fourier transforms, the function F is entire and not identically zero; it is also bounded on the upper half plane, since kf (t)eitzk_[0,H] ≤ eH_{kf (t)k}

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Imz > 0. Then, G is also not identically zero, is bounded in the open disk ∆, and is continuous on ¯∆, except for the point w = −1. Now by a consequence of Jensen’s formula, see [44, p. 307], we have

1 2π

π

Z

−π

log |G(eiϑ)|dϑ > −∞.

By applying a change of variable x = i−ie_1−eiϑiϑ = 2 tan(ϑ/2), which changes the differentials

by dϑ = _1+r2dr2, we have 1 π +∞ Z −∞ log |F (r)| 1 + r2 dr > −∞.

By partial integration of F we have

F (z) = H Z 0 f (t)eitzdt = 1 (iz)k H Z 0 f(k)(t)eitzdt,

since all boundary terms vanish, thanks to f being compactly supported. Then by the bounds kf(k)k_[0,A] ≤ 2−kk!mk on the derivatives of f , we have

|rkF (r)| ≤ 2kHk!mk for r ∈ R and k ∈ N. Hence S(r)|f (r)| = |f (r)|max k≤r rk k!mk ≤ ∞ X k=0 rk|f (r)| k!mk ≤ 2H for r > 0

from which we have

∞ Z 1 log S(r) 1 + r2 dr = ∞ Z 1 log(S(r)|f (r)| 1 + r2 dr − ∞ Z 1 log |f (r)| 1 + r2 dr < +∞; .

Proof that (iii) implies (ii). Let us denote by µ = (µk)0<k∈N the sequence µk =

Mk/Mk−1 = kmk/mk−1, as in section 2. We define the function

M_{(r) = #{k ∈ N|} r µk

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One can easily verify that S(r) = Q

r µk>1 r µk, hence log S(r) = X r µk>1 log( r µk ) ≥ X r µk>e log( r µk ) ≥ M(r).

Now, on the one hand we have

ej+1 Z ej log S(r) 1 + r2 dr ≥ M(ej) 1 + e2j+2 ej+1 Z ej dx ≥ 1 2e2 M(ej) ej ; (1.4.1)

on the other hand instead

X

ej−2_<µ k<ej−1

1 µk

≤ (M(ej) − M(ej+1))e2−j ≤ e2M(e

j₎

ej . (1.4.2)

Summing 1.4.1 and 1.4.2 over j we obtain

X M_k Mk+1 ≤ 2e4 ∞ Z e2 log S(r) 1 + r2 dr.

Proof that (i) implies (iii). Let µk be as above and assume that

P ₁

µk < ∞. The

claim is that the infinite product

f (z) = sin z z 2 ∞ Y k=1 sin_µz k z µk

defines an entire function of exponential type which is not identically zero and which satisfies the inequalities

|xhf (x)| ≤ h!mh

sin x x

2 for x real.

To start, we will show that the infinite product is, in fact, well defined. The function 1 − (sin z)/z has a zero at the origin, so that we can find a constant B > 0 such that

1 −sin z z ≤ B|z| for |z| ≤ 1

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and that 1 −sin z µk z µk ≤ B|z| µk for |z| ≤ µk,

so that the sum

∞ X k=1 1 −sin z µk z µk

converges on compact sets. The infinite product above defines an entire function f (z), which is not identically zero.

From the identity

sin z z = 1 2 1 Z −1 eitzdt we have sin z z ≤ e|z| and |f (z)| ≤ eC|z| where C = 2 + ∞ X k=1 1 µk .

Now, if x is real, from the inequalities | sin x| ≤ |x| and | sin x| ≤ 1, we have

|xhf (x)| = |xh| sin x x 2 ∞ Y k=1 sin_µx k x µk ≤ |xh| sin x x 2 h Y k=1 sin_µx k x µk ≤ ≤ |xh| sin x x 2 h Y k=1 µk x = h!mh sin x x 2

and by integrating this inequality we obtain

1 π +∞ Z −∞ |xk_{f (x)|dx ≤ k!m} k

The function f then falls under the hypotheses of the Paley-Wiener theorem, see [44, p. 375] which affirms that the Fourier transform F of f

F (t) = 1 2π +∞ Z −∞ f (x)e−itxdx

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theorem for the derivatives of the Fourier transform

F(k)(t) = 1 2π

+∞

Z

−∞

(−ix)kf (x)e−itxdx which, using the inequality we previously proved gives

kF(k)_{k ≤} 1 2π +∞ Z −∞ |xk_{f (x)|dx ≤} 1 2k!mk.

Hence the function F is a compactly supported function of class E1(m) and the class

E1(m) is not quasi-analytic.

As we announced at the beginning of this section, we will now show a proof of the theorem of Carleman [11, 13] on the failure of surjectivity of the Taylor series map ˆ· : En(m) → Fn(m) in the strictly quasi-analytic case.

Theorem 1.20. If m = (mk)k∈N is a logarithmically convex sequence such that O (

E_n(m) and En(m) is quasi-analytic, then the map

ˆ· : En(m) → Fn(m),

that associates to a function in En(m) its Taylor series, is not surjective.

We will need to define some Banach spaces before proceeding with the proof. For every integer ν > 0 let Iν be the real interval (−1/ν, 1/ν) and for every real σ > 0 and

f ∈ C∞(I_νn) we put kf k_n,ν,σ = sup α∈Nn kDα_{f k} ∞ σ|α|_α!m |α|

where k · k∞ is the usual L∞ norm on Iνn. Put En,ν,σ(m) = {f ∈ C∞(I)|kf kn,ν,σ < ∞}.

It is easy to verify that (En,ν,σ(m), k ˙kn,ν,σ) is a Banach space.

Let C∞( ¯I) denote the space of infinitely differentiable functions on the interval I and whose derivatives of every order extend continuously to ¯I. If k ˙k2 is the usual L2 norm on

I, we define the norm

kuk2 ₌ ∞ X k=0 ku(k)_k2 2 (k!mk)2

and set H(m) = {u ∈ C∞( ¯I)|kuk < ∞}. Then (H(m), k ˙k) is a Hilbert space, whose associated scalar product will be denoted h·,·i.

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For a function u ∈ C∞( ¯I), we have the inequalities 1 √ 2ku (k)_k 2≤ ku(k)k∞≤ √ 2 ku(k)k2+ ku(k+1)k2 .

Using these inequalities we have, for η ∈ (0, 1), the continuous inclusions

E1,1,1−η(m) ⊆ H(m) ⊆ E1,1,1+η(m+)

where the sequence m+ is the shifted sequence defined by m+_k = mk+1 for all k ∈ N. We

remark that the inclusions are in fact compact, see [31].

The proof we expose is the one due to Thilliez [47], which avoids the subtle variational arguments from Carleman’s original proof with the use of Banach and Hilbert spaces.

Proof. As was the case for the Denjoy-Carleman theorem, it is enough to prove this theorem in the 1-dimensional setting.

Consider, given a function g ∈ H(m), the system of equations

v(j)(0) = g(j)(0) for 0 ≤ j < k. (1.4.3) For each j ∈ N the map u → u(j)(0) is a continuous linear functional on H(m) and, by the Riesz representation theorem, there is an element ej ∈ H(m) such that

u(j)(0) = hej, ui for all u ∈ H(m).

Denote by Vk the span of e0, e1, . . . , ek−1. If we assume that v is an element of Vk, we

can write v =P λjej and the system of equations 1.4.3 can be written in terms of the

coefficients λj as

k−1

X

i=0

λihei, eji = g(j)(0). (1.4.4)

One can verify that the ej are linearly independent by checking the scalar products with

the powers of x. The matrix (hei, eji) is then invertible and the system 1.4.4 admits a

solution gk=P λj,kej. This solution depends linearly on the first k − 1 derivatives of g

and we can write

gk(t) = k−1

X

j=0

uj,k(t)g(j)(0), (1.4.5)

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minimal in H(m). Let u be another solution of the system, the function u must satisfy

he_j, u − gki = uj(0) − g_k(j)(0) = 0 for 0 ≤ j < k

so that u − gk∈ Vk⊥ and, since gk∈ Vk, we have

kuk2_{= kg}

kk2+ ku − gkk2 ≥ kgkk2.

Thie minimality of the solution gk implies, in particular, that kgkk ≤ kgk so that

the sequence (gk)k∈N is bounded in H(m). The claim is now that (gk) converges in

E1,1,1+η(m+). Since, as we have remarked, the inclusion H(m) ⊆ E1,1,1+η(m+) is

compact, we only have to show that g is the only possible limit for (gk). Let h be another

accumulation point for the sequence (gk), by continuity we have, for all j ≥ 0,

h(j)(0) = lim

p→∞g (j)

kp(0) = g

(j)₍₀₎

and we can conclude h = g by the quasi-analyticity of the class E1(m).

Let f be an element of E1(m); for an x sufficiently close to 0, the function ˜f (t) = f (xt)

is in the space E1,1,1−η(m) and 1.4.5 applied for t = 0 and putting ωj,k= j!uj,k(1) gives

f (x) = lim k→∞ k−1 X j=0 ωj,k f(j)(0) j! x j _(1.4.6)

for x sufficiently close to 0.

If we apply formula 1.4.6 to the cases f (x) = xj for j ≤ 0, we obtain xj = lim k→∞ k−1 X j=0 ωj,k (xj)(j)_x=0 j! x j _{= lim} k→∞ωj,kx j_,

which gives limk→∞ωj,k= 1. We can then select a subsequence (ωj,kp) of (ωj,k) such that

kp−1

X

j=1

|1 − ω_j,k_p|mj ≤ 1 (1.4.7)

for all p ≤ 0.

Finally, consider the formal series F =P Fjxj where Fj = mj when j = kp for some

p ≤ 0 and Fj = 0 otherwise. We claim that F is not the Taylor series at 0 of any function

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all j ≤ 0. We then have f (x) = lim p→∞ kp−1 X j=0 ωj,kpFjx j_.

Thanks to how the sequence (Fj) was defined, we have kp−1 X j=0 ωj,kpFjx j ₌ kp−1 X j=0 ωj,kpFjx j and kp−1 X j=0 Fjxj = p−1 X q=0 mkqx kq_,

so that we can write

f (x) = lim p→∞   p−1 X q=0 mkqx kq kp−1 X j=0 (ωj,kp− 1)Fjx j  .

We have chosen the subsequence kp so that the second sum in the equation is bounded by

1 thanks to 1.4.7, from which we have that the first sum must also be bounded. But the class E1(m) strictly contains the analytic class O, which means that limk→∞(mk)1/k = ∞

so that the first sum cannot be bounded; as such we have reached a contradiction and the proof is finished.

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Chapter 2

Weierstrass division and

preparation

We have seen in the first chapter how the rings En(m) relate to rings On and Fn. A

fundamental classical result for these latter two rings is that they are noetherian rings. This is usually proved through the Weierstrass division and Weierstrass preparation theorems. As such, whether some form of Weierstrass division and preparation holds for Denjoy-Carleman classes has been a question of interest. Childress [18] proved that, in general, Weierstrass division fails in a Denjoy-Carleman class, when n ≥ 2, and what properties must a quasi-analytic function f have for division by f to be possible. A positive result for classes Fn(m) formal series was reached by Chaumat and Chollet [16],

who showed that Weierstrass division is possible for a wide number of series and that Fn(m) is consequently noetherian.

Our purpose in this chapter is to review the main results on the failure of Weier-strass division and preparation in Denjoy-Carleman classes, starting from Childress and examining the later developments.

2.1 Weierstrass division and hyperbolic polynomials

As we will need to single out a variable, it will be practical to write, throughout this chapter, the coordinates of Rn as (x, t) = (x1, . . . , xn−1, t) and the coordinates of Rn−1

as x = (x1, . . . , xn−1).

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Definition 2.1. A function f ∈ En(m) is called regular of order d with respect to t if

∂k_f

∂tk(0) = 0 for k < d and

∂d_f

∂td(0) 6= 0

or, equivalently, if there exists a unit u ∈ E1(m) such that

f (0, . . . , 0, t) = u(t)td.

Definition 2.2 (Weierstrass division property). A function f ∈ En(m), regular of order

d with respect to t, is said to have the Weierstrass division property if, for every function g ∈ En(m), there exist functions q ∈ En(m) and r1, . . . , rd∈ En−1(m) such that

g(x, t) = q(x, t)f (x, t) + r1(x)td−1+ r2td−2+ · · · + rd(x). (2.1.1)

The sum r1(x)td−1+ · · · + rd(x) will sometimes be denoted as r(x, t).

Definition 2.3 (Weierstrass preparation property). A function f ∈ En(m), regular of

order d with respect to t, is said to have the Weierstrass preparation property if there exist functions u ∈ En(m) a0, . . . , ad−1 ∈ En−1(m), with u(0) 6= 0 and aj(0) = 0 for

j = 1, . . . , d, such that

f (x, t) = td+ a1(x)td−1+ · · · + ad(x)u(x, t). (2.1.2)

A function of the form f (x, t) = td+a1(x)td−1+· · ·+ad(x), with aj(0) = 0 for j = 1, . . . , d,

is called a Weierstrass polynomial or a distinguished polynomial.

We say that the ring En(m) has the Weierstrass division (resp. preparation) property

if every function f ∈ En(m) has it.

The Weierstrass division property for a function f ∈ En(m) implies the preparation

property for f as in the classical version of the theorems. The proof is exactly the same: if f is regular of order d in t, divide td by f to obtain

td= q(x, t)f (x, t) +

d

X

j=1

rj(x)td−j.

Considering the derivatives in t of order up to d of both terms it is easy to see that q must be a unit in En(m) and that rj must belong to the maximal ideal of En−1(m) for

j = 1, . . . , d.

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2.1. WEIERSTRASS DIVISION AND HYPERBOLIC POLYNOMIALS 39

quasi-analytic class with the Weierstrass division property is the analytic class. We need two lemmas before showing Childress’s example. The second lemma in particular is of utility when studying the Weierstrass division property in quasi-analytic classes.

We recall two constructions from the previous chapter that are used in the lemmas. The function S(r) is defined by S(r) = sup

k≤r

rk/(k!mk) . The norm k · kn,ν,σ for integers ν, σ > 0

is defined for C∞functions on In

ν = (−1/ν, 1/ν)nby kf kn,ν,σ = sup α∈Nn

kDα_{f k}

∞/(σ|α|α!m|α|

and its associated Banach space is En,ν,σ(m).

Lemma 2.4. If En(m) is a Denjoy-Carleman class and there exist positive real constants

ε, A and C such that

eεr≤ C · S(r) for all r > a, then En(m) = On. Proof. We have max k≤r ekrk k! ≤ e εr

and, if we look at the proof of lemma 1.8, we see that there exists k0∈ N such that

mk≤ C(ε−1)k for all k ≥ k0.

By theorem 1.3 we know that this means that En(m) = On.

Lemma 2.5. Consider f ∈ En(m) with the Weierstrass division property. Let g ∈ En(m)

and q ∈ En(m), rj ∈ En−1(m), j = 1, . . . , d, such that

g(x, t) = q(x, t)f (x, t) +

d

X

j=1

rj(x)td−j.

If ν0 ∈ N is such that f is defined on Iνn0 = (−1/ν0, 1/ν0)

n _{and ν, σ > 0 with ν ≥ ν} 0

are such that g is defined on I_νn = (−1/ν, ) and kgkm,ν,σ < ∞, then there exist C > 0

and ν0, σ0 > 0 with ν0 ≥ ν0 and such that q is defined on I_νn0, rj is defined on I_νn−10 for

j = 1, dots, d, and

kqk_n,ν0_,σ0 < Ckgk_n,ν,σ and

kr_kk_n−1,ν0_,σ0 < Ckgk_n,ν,σ for j = 1, . . . , d.

Proof. When σ0 < σ we have continuous inclusions En,ν,σ0(m) → E_n,ν,σ(m) and when

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that these two families of maps commute. It is then well-defined the direct limit of Banach spaces lim_−→En,ν,σ(m), where the limit is taken with respect to both ν and σ. We

define the map

A : E = lim_−→ En,ν,σ(m) ⊕ En−1,ν,σ,(m) d → lim_−→En,ν,σ(m) = F (q, r1, . . . , rd) 7→ qf + r = g

The claim is that this map is an isomorphism.

The map A is continuous since each restricted map from En,ν,σ(m) ⊕ (En−1,ν,σ(m))d)

to F is continuous, so that A also is, by the properties of the direct limit. The map A is injective because the class En(m) is quasi-analytic and is surjective because f has the

Weierstrass division property. That the inverse map of A, which we will call T , is also continuous is a consequence of a Grothendieck version of the open mapping theorem [27].

Since T is continuous, we have that, if ν, ν0 ≥ ν0 and σ, σ0 > 0 are such that

g ∈ En,ν,σ(m) and T g ∈ En,ν0_,σ0(m) ⊕ (E_n−1,ν0_,σ0(m))d, there is a constant C > 0 such

that

kT gk_ν0_,σ0 ≤ Ckgk_n,ν,σ,

which immediately gives

kqk_n,ν0_,σ0 ≤ Ckgk_n,ν,σ,

kr_kk_n−1,ν0_,σ0 ≤ Ckgk_n,ν,σ.

We now show Childress’s counter-example.

Proposition 2.6. If En(m) is a Denjoy-Carleman class, with n ≥ 2, that has the

Weierstrass division property, then En(m) = On.

Proof. We will limit ourselves to the case n = 2. In the higher dimensions, one can simply consider the same functions used for the 2-dimensional case, viewing them as constant along the other variables. We choose functions f (x, t) = t2_{+ x}2 _{and g(t, a) = e}iat_{, where}

a ∈ R is a parameter. Clearly f, g ∈ E2(m) and f is regular of order 2 in t, so that we

can write

eiat= q(x, t, a)(t2+ x2) + t · r1(x, a) + r2(x, a). (2.1.3)

with qa∈ E2(m), ra,1, ra,2∈ E1(m). Note that, since f and g are both entire functions, we

can, for each a ∈ R find analytic functions q, r1, and r2 that give ??, without resorting

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The norm k · k2,1,1 of g(t, a) for fixed a is

keiatk_2,1,1= sup α∈N2 k|Dα_eiat_k ∞ α!m|α| = sup k∈N k(ia)k_eiat_k ∞ k!mk = sup k∈N ak k!mk = S(a) (2.1.4)

At this point, we use the hypothesis of f having the Weierstrass division property to apply lemma 2.5. The lemma gives us constants ν > 0, σ > 0, K > 0, independent of a, such that

|r1(x, a)| ≤ kr1(·, a)k1,ν,σ ≤ Kkg(·, a)k2,1,1= K · S(a) (2.1.5)

for all x ∈ Iν.

Since all functions involved are analytic, it makes sense to evaluate equation 2.1.3 at the roots of f , that is t = ±ix, which yields

e−ax= ixr1(x, a) + r2(x, a)

and

eax= −ixr1(x, a) + r2(x, a).

Subtracting the first from the second we get

r1(x, a) = i(eax− e−ax)/2x.

If we choose a positive ε < 1/ν and evaluate for x = ε, from 2.1.5 we have

|eεa_{− e}−εa_{|/2ε = |r}

1(ε, a)| ≤ K · S(a).

For a 0 we have (eεa− e−εa)/2ε ≈ eε/2ε so that we can find real constants A, C > 0 such that eaε≤ C · S(a) for a > A. Lemma 2.4 now gives En(m) = On.

We stress again that the crucial point, where the Weierstrass division property was used in Childress’s example, was in applying lemma 2.5 and not in finding the functions q, r1, r2, which can be obtained without much problem from hyperbolic sine and cosine.

Even though we can not hope for the Weierstrass division property to hold for the whole ring En(m), we are still interested in what properties must an element f ∈ En(m)

have so that f has the division property.

It turns out, as we will see shortly, that the property of (t2+ x2) that allowed the counter-example to work is that it has non-real roots. With this in mind we give the following

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Definition 2.7. We say that a Weierstrass polynomial P (x, t) ∈ En−1(m)[t] is a

hyper-bolic polynomial if there exist a neighbourhood Ω of the origin in Rn−1 such that P is defined on Ω × C and, for all x ∈ Ω, the polynomial P (x, z) ∈ C[z] has only real roots.

It is a theorem of Childress [18] that if a function f ∈ En(m), with n ≥ 2, has the

Weierstrass division property, then f is a hyperbolic polynomial, up to units of En(m).

Theorem 2.8. If En(m) ) On is a Denjoy-Carleman class, with n ≥ 2, and f ∈ En(m)

has the Weierstrass division property, then there exist a hyperbolic polynomial F (x, t) = td_{+ a}

1(x)td−1+ · · · + ad(x) ∈ En−1(m)[t] and a unit u(x, t) ∈ En(m) such that f = uF .

The proof we follow is not the original one of Childress, but a shorter one due to Chaumat and Chollet [17].

Proof. We will assume, without loss of generality, that f is already a distinguished polynomial, that is f = F , and go on to show the claim about the roots of F .

Consider a one variable polynomial G ∈ C[z]. The polynomial G defines a germ g ∈ En(m) by g(x, t) = G(t). Applying the Weierstrass division property we have

G(t) = g(x, t) = q(x, t)F (x, t) +

d

X

j=1

rj(x)td−j.

If we denote by P (λ, z) = zd+ λ1zd−1+ · · · + λd∈ C[z] a generic polynomial of degree

d, with λ ∈ Cd_{, by euclidean division, we have}

??G(z) = Q(λ, z)P (λ, z) +

d

X

j=1

Rj(λ)zd−j (2.1.6)

where Q(λ, z) and Rj(λ) depend polynomially from λ and z. For a sufficiently small

neighbourhood Ω of the origin in Rn−1 _{we evaluate equation ?? for λ = a(x) =}

(a1(x), . . . , ad(x)), where the aj(x) are the coefficients of the Weierstrass polynomial

F . This gives us G(z) = Q(a(x), z)F (z) + d X k=1 Rj(a(x))zd−j.

By quasi-analyticity of the class En(m) we must have

Q(a(x), t) = q(x, t) and Rj(a(x)) = r(x) for all j = 1, . . . , d.

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not hyperbolic. Then, there exists a point (x0, z0) ∈ Ω × (C \ R) such that F (x0, z0) = 0

and G(z0) = d X j=1 rj(x0)z0d−j.

Choose integers ν, σ > 0, surely kgkn,ν,σ, since g is a polynomial. By lemma 2.5 we know

that krjkn−1,ν0_,σ0 ≤ Ckgk_n,ν,σ for all j = 1, . . . , d and some C > 0 and ν0, σ0 > 0. This

gives us |G(z0)| ≤ Ckgkn,ν,σ d X j=1 |j0|d−j;

our claim is that such an inequality is impossible when z0 has non-zero imaginary part.

Let δ = | Im z0|/2 and V = {z ∈ C| d(z, [−1/ν, 1/ν]) < δ}. For each t ∈ Iν, by

Cauchy’s integral formula we have

|G(k)_{(t)| ≤ k!δ}k_sup z∈ ¯V

|G(z)|.

From this we have

kgkn,ν,σ = sup α∈Nn kDα_gk ∞ σ|α|α!m|α| = sup k∈N kg(k)_k ∞ σk_k!m k ≤ sup k∈N δksup |G(z)| σk_m k .

Since the class En(m) strictly contains the analytic class, we have lim k→∞(mk) 1/k _{= ∞ by} lemma ??, so that kgkn,ν,σ ≤ Dsup z∈ ¯V |G(z)|.

Now, the set K = ¯V ∪ {z0} is compact and C \ K is connected, so that, by Runge’s

approximation theorem, we can find a sequence of polynomials Gh with

Gh(z0) = 1 and sup z∈ ¯V |G_h(z)| ≤ 2−h, so that we have 1 = Gh(z0) ≤ D2−h d X j=1 zd−j₀ → 0 which is absurd.

A converse of this theorem was proved by Chaumat and Chollet [17], so that we can state

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Theorem 2.9. A Weierstrass polynomial P (x, t) in a quasi-analytic class En(m) with

n ≥ 2 has the Weierstrass division property if and only if it is hyperbolic.

2.2 Later proofs

We now also want to show another, more recent, proof of the failure of the Weierstrass division theorem, due to Elkhadiri and Sfouli [24]. Their proof applies to the more general setting of quasi-analytic systems, rather than only to the setting of Denjoy-Carleman classes that we are considering here, and as such can be considered a generalization of Childress’s result. In the author’s opinion the proof is also quite ingenious in its brevity. We will also show a proof of the failure of the Weierstrass preparation property for quasi-analytic classes, due to Parusi´nski and Rolin [41] which has much of the same flavor as Elkhadiri and Sfouli’s proof.

Lemma 2.10. If F is a formal series in Fnand for each unit length vector ξ ∈ Sn−1⊆ Rn

we have that the formal series Fξ(t) = F (ξt) is convergent, then F is convergent.

Proof. See [1].

Theorem 2.11. If En(m) is a Denjoy-Carleman class, with n ≥ 3, that has the

Weier-strass division property, then En(m) = On.

Proof. We will show that, if the division property holds for E3(m), then E1(m) equals

the analytic class O1. From this the full result can be recovered thanks to lemma ??

Let us denote the coordinates of R3 by (x, y, t). Let f (x) ∈ E1(m) and put g(x, t) =

f (x + t). Consider (y2+ t2), which is regular in t of order 2, and divide g by it, to obtain g(x, t) = (t2+ y2)q(x, y, t) + t · r1(x, y) + r2(x, y).

From this identity, by passing to the respective Taylor series, we have

ˆ

g(x, t) = (t2+ y2)ˆq(x, y, t) + t · ˆr1(x, y) + ˆr2(x, y).

If we now evaluate the power series for t = iy, we obtain

ˆ

f (x + iy) = ˆr2(x, y) + iyˆr1(x, y).

Let us denote u(x, y) = r2(x, y) and v(x, y) = yr1(x, y). We have

f0(x) = ∂g ∂x(x, 0) = ∂r2 ∂x(x, 0) = ∂u ∂x(x, 0)

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2.2. LATER PROOFS 45

so that, if u is analytic, f is too. To show that u is analytic we will prove that it is the real part of an holomorphic function.

Evaluating the derivatives of ˆf (x + iy) with respect to x and y we get

∂ ˆf (x + iy) ∂x = ∂ ˆu ∂x(x, y) + i ∂ ˆv ∂x(x, y) and ∂ ˆf (x + iy) ∂y = ∂ ˆu ∂y(x, y) + i ∂ ˆv ∂y(x, y). By the chain rule we have

∂ ˆf (x + iy) ∂y = i ˆf 0_{(x + iy) = i}∂ ˆf (x + iy) ∂x , which gives 0 = ∂ ˆf (x + iy) ∂x + i ∂ ˆf (x + iy) ∂y = ∂ û ∂x(x, y) + i ∂ ˆv ∂x(x, y) + i ∂ û ∂y(x, y) − ∂ ˆv ∂y(x, y). We then get ∂ û ∂x = ∂ ˆv ∂y and ∂ û ∂y = − ∂ ˆv ∂x

which are exactly the Cauchy-Riemann equations for ˆf (x + iy). This means that the function f (x + iy) is holomorphic and u(x, y) is analytic as we wanted.

As anticipated, we now show a result of Parusi´nski and Rolin [41] that proves that a Denjoy-Carleman class with the Weierstrass preparation property must be equal to the analytic class. This is of interest, as the preparation property is a priori weaker than the division property.

Lemma 2.12. Let f ∈ En(m) such that f (0, t) = t2+ t3+ h(t) where h(t) is of order

strictly greater than 3 in t. If the Weierstrass preparation theorem holds for En(m), then

there exists f1, f2 ∈ En(m) such that

f (x, t) = tf1(x, t2) + f2(x, t2).

Proof. We define functions g1(x, t) = (f (x, t) − f (x, −t))/2 and g2(x, t) = (f (x, t) +

On quasi-analytic Denjoy-Carleman classes

Universit`

a degli Studi di Pisa

Dipartimento di Matematica

Corso di Laurea Magistrale in Matematica

Tesi di Laurea Magistrale

On quasi-analytic

Denjoy-Carleman classes

12 maggio 2017

Candidato

Lapo Dini

Prof.ssa Francesca Acquistapace

Relatore

Abstract

Contents

Introduction

Chapter 1

Denjoy-Carleman classes

1.1

Ultra-differentiable functions

1.2

Inclusions between classes of ultra-differentiable

func-tions

1.3

Composition, inverses and derivation

1.4

The Denjoy-Carleman theorem

Chapter 2

Weierstrass division and

preparation

2.1

Weierstrass division and hyperbolic polynomials

2.2

Later proofs