PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 247–262 DOI: https://doi.org/10.2298/PIM1716247Z

HOLOMORPHIC SERIES EXPANSION OF FUNCTIONS OF CARLEMAN TYPE ON THE INTERVAL [−1, 1]

Hicham Zoubeir

Abstract. We characterize the functions of some Carleman classes on the unit interval [−1, 1] as sums of holomorphic functions in specific neighborhoods of [−1, 1]. As an application of our main theorem, we perform an alternative construction of the Dyn’kin’s pseudoanalytic extension for these Carleman classes on [−1, 1].

1. Introduction

In 1926 [8] Carleman raised the problem of the representation of the functions of a quasianalytic class in terms of their successive derivatives at a given point. He noticed that this problem should be solved by a decomposition method. This problem was also raised by Julia in 1925 [13, 14, 15], while he was looking for an algorithmic generalization of the classical Borel process which generates classes of quasi-analytic functions from sequences of complex numbers converging to 0. In 1962 [2] Badalyan gave, by his theory of quasi-powers (a generalization to quasianalytic Carleman classes of Taylor series expansion) the complete solution to Carleman’s problem. In 1970 [3] Badalyan generalized his theory to some nonquasianalytic classes. In 1991 [10, pp. 249–253] Ecalle obtained for the functions of a regular Carleman class on a segment [a, b], a series expansion into holomorphic functions on specific neighborhoods of [a, b]. In 2004, Belghiti obtained for certain Carleman classes on arbitrary bounded convex planar domains [4] a similar but more explicit holomorphic expansion series. Let us observe that the approach in [4, 10] relies mainly on the theorem of pseudoanalytic extension due to Dyn’kin [9].

Improving the methods of Ecalle and Belghiti, we obtained in [5] a characterization of the functions of a Gevrey class on [−1, 1] as sums of series of holomorphic functions in suitable neighborhoods of [−1, 1], and here we generalize this method to some Carleman classes on [−1, 1]. As an application of our main theorem, we derive an alternative construction of Dyn’kin’s pseudoanalytic extension for these Carleman classes.

2010 Mathematics Subject Classification: Primary 30D60; Secondary 41A20.

Key words and phrases: Carleman classes.

Communicated by Stevan Pilipović.

2. Preliminary notes

Let S be a nonempty subset of ℂ, f: S → ℂ a bounded function, and ∥f∥_{∞,S} := sup_{z∈S} |f(z)|. For z ∈ ℂ we set ρ(z, S) := inf_{u∈S} |z − u|. For r > 0, B(z, r) is the usual open ball in ℂ with center z and radius r. We set also

S_{r} := S + B (0; r) := {z + u : z ∈ S, u ∈ B(0, r)}

Thus we have S_{r} = {z ∈ ℂ : ρ(z, S) < r}. O(S) denotes the set of holomorphic functions on some neighborhood of S.

Let α := (α_{1}, α_{2}), β := (β_{1}, β_{2}) ∈ ℕ^{2}. We write β ⩽ α if: β_{1} ⩽ α_{1} and β_{2} ⩽ α_{2}. Given a property 𝔓(x), with x ∈ ℝ, we say that 𝔓(x) holds ultimately if there exists a_{0} ∈ ℝ such that 𝔓(x) holds for all x ⩾ a_{0}. We define an equivalence relation on the set F of real valued functions which are defined on a real half-line by writing f =_{∞} g if we have f(x) = g(x) ultimately. We denote by [f] the class of equivalence of a function f ∈ F for the equivalence relation =_{∞} . The quotient set G := F/=_{∞} is endowed with operations of addition and multiplication induced by those of F making G into a commutative ring. The classes of equivalence for this relations are called the germs of functions at +∞. To simplify the writing we will identify the germ [f] with its representant f. We consider the field ℝ of real numbers as a subring of G, by identifying a real number a with the germ of the function x ↦ a.

We denote by G_{1} the subring of G consisting of the germs of functions that are ultimately of class C^{1}. A subring ℵ of G_{1} is called a Hardy field if ℵ is a field which is stable by derivation. Functions belonging to a Hardy field ℵ have the following properties: they are ultimately strictly monotone unless they are ultimately constant, they are ultimately of constant sign unless they are ultimately identically vanishing. It follows that for every f ∈ ℵ the limit lim_{x→+∞} f(x) exists in ℝ ∪ {+∞, −∞}, and that for every f, g ∈ ℵ we have ultimately one of the following cases f(x) < g(x), g(x) < f(x), f(x) = g(x).

We say that an element f of ℵ is bounded if lim_{x→+∞} f(x) ∈ ℝ, infinitesimal if lim_{x→+∞} f(x) = 0, and infinite if lim_{x→+∞} |f(x)| = +∞.

If f, g ∈ ℵ and g is infinite and ultimately positive, then f ∘ g ∈ G_{1} is by definition the germ in G_{1} such that ultimately (f ∘ g )(x) = f(g(x)).

In our work we will need the following results.

Theorem 2.1. [1, 19]. Let f be an infinite and ultimately positive element of a Hardy field ℵ. Then there exists a Hardy field H and a germ g ∈ H such that g is an infinite and ultimately positive element of the Hardy field H and (f ∘ g)(x) = x ultimately. g is called the inverse of f and is denoted by g^{⟨−1⟩} .

Theorem 2.2. [1, 18, 20]. Let F (Y), G(Y) ∈ ℵ[Y] and y ∈ G_{1} be such that G(y) ≠ 0 and y′ G(y) = F (y) (in G_{1}). Then the ring of germs ℵ[y] is an integral domain with fraction field ℵ(y) ⊂ G_{1}, and ℵ(y) is a Hardy field.

As a consequence of this theorem, it follows that a Hardy field ℵ can be enlarged to a Hardy field ℵ_{0} containing the germ Id of the identity function and the germ ln of the logarithmic function. The following theorem provides a strong generalization of this remark.

Theorem 2.3. [7] Let ℵ be a Hardy field. There exists a Hardy field ℵ_{1} containing ℵ such that the germs at +∞ of the functions exp∘f, ln∘|f| belong to ℵ_{1} for every function f ∈ ℵ_{1} which is not ultimately identically vanishing.

A positive function measurable fuction f defined on some neighborhood of +∞ is said to be regularly varying with index τ ∈ ℝ if lim_{x→+∞} \frac{f(Cx)}{f(x)} = C^{τ}, C > 0. We set ℑ(f) := τ. If ℑ(f) = 0, then we will say that the function f is slowly varying.

If f is regularly varying with index τ, then there exists a slowly varying function L such that for f(x) = x^{τ}L(x), for sufficiently large values of x.

Let f be a function defined on an interval of the form [a, +∞[ such that f is strictly positive and belongs as a germ at +∞ to a Hardy field. Then according to [12], the function f is regularly varying if and only if lim_{x→+∞} \frac{xf′(x)}{f(x)} ∈ ℝ. Then we have ℑ(f) = lim_{x→+∞} \frac{xf′(x)}{f(x)} .

Theorem 2.4 (Potter’s bounds, [6]). Let f be a regularly varying function of index τ. For every ε > 0, we have ultimately (1 − ε)x^{τ−ε} ⩽ f(x) ⩽ (1 + ε)x^{τ+ε} .

Let μ: ℝ_{+}^{∗} → ℝ be a function of class C^{2} on ℝ_{∗}^{+} which belongs, as a germ at +∞, to a Hardy field ℵ containing the germ at +∞ of the function x ↦ lnx. Since the function μ belongs as a germ at +∞ to the Hardy field ℵ, it follows that the limit σ(μ) := lim_{t→+∞} \frac{ln(t)}{μ(t)} exists in ℝ_{+} ∪ \{+∞\}. σ(μ) is called the order of the function μ. We assume that 0 < σ(μ) < +∞. It follows then that we have

lim_{t→+∞} μ(t) = +∞, μ(t) = O_{t→+∞} (t)

Furthermore, we have by virtue of L’Hopital’s rule, lim_{t→+∞} tμ′(t) = \frac{1}{σ(μ)} . Thence we have lim_{t→+∞} \frac{tμ′(t)}{μ(t)} = 0. Consequently the function μ is slowly varying.

Consider the function M_{μ} defined on ]0, +∞[ by M_{μ}(t) := t^{t}e^{tμ(t)} , t > 0. The functions Ω_{μ} and H_{μ} are defined on ℝ_{+}^{∗} by

Ω_{μ}(x) := inf_{t>0} [ \frac{M_{μ}(t)}{x^{t}} ], x > 0, H_{μ} (x) = inf_{t>0} [ \frac{M_{μ}(t)}{t^{t}x^{t}} ], x > 0

We consider also the sequence M_{μ} := (M_{n})_{n∈ℕ^{∗}} defined by M_{n} := M_{μ}(n), n ∈ ℕ^{∗}.

Let W be a nontrivial interval of ℝ. The Carleman class C_{M_{μ}} (W) is the set of functions f of class C^{∞} on W such that sup_{x∈W} |f^{(n)} (x)| ⩽ Cρ^{n}M_{n}, n ∈ ℕ^{∗} where C, ρ > 0 are real constants.

We denote by Λ_{M_{μ}} the set of sequences (a_{n})_{n∈ℕ} of complex numbers such that |a_{n}| ⩽ Cρ^{n}M_{n}, n ∈ ℕ^{∗} where C, ρ > 0 are constants. We denote by ω_{μ} and h_{μ} the functions defined by ω_{μ}(x) := − ln[Ω_{μ}(x)] and h_{μ}(x) := − ln[H_{μ}(x)], respectively.

Let γ_{μ} denote the function ultimately defined by the system

(2.1)  x = t^{2}μ′(t), γ_{μ}(x) = μ(t) + tμ′(t),

the parameter t being uniquely determined by x. We denote then t by t_{0}(x).

Let φ_{μ} denote the function defined by φ_{μ}(x) := ω_{μ}(x) − xω′_{μ}(x) for sufficiently large values of x.

The following propositions will play a crucial role in the proof of our main result.

Proposition 2.1. 1. The function ω_{μ} is ultimately well defined by the system

(2.2)  x = et exp[μ(t) + tμ′(t)], ω_{μ}(x) = t + t^{2}μ′(t)

the parameter t being ultimately uniquely determined by x. We denote then t by t_{1}(x).

2. The function ω_{μ} is ultimately strictly concave.

3. The function φ_{μ} is ultimately well defined by the system

(2.3)  x = et exp[μ(t) + tμ′(t)], φ_{μ}(x) = t^{2}μ′(t)

the parameter t being ultimately uniquely determined by x.

4. The function φ_{μ} is an increasing diffeomorphism between neighborhoods of +∞. The inverse function N_{μ} := φ_{μ}^{⟨−1⟩} is ultimately defined by the system

(2.4)  x = t^{2}μ′(t), N_{μ}(x) = et exp[μ(t) + tμ′(t)]

the parameter t being ultimately uniquely determined by x 5. The function h_{μ} is ultimately well defined by the system

(2.5)  x = exp[μ(t) + tμ′(t)], h_{μ}(x) = t^{2}μ′(t)

the parameter t being utimatey uniquely determined by x. Furthermore h_{μ} is ultimately positive and infinite so it has an inverse h_{μ}^{⟨−1⟩} which belongs to a Hardy field.

6. Each of the function ω_{μ}, φ_{μ}, h_{μ}, N_{μ}, γ_{μ} belongs to a Hardy field.

7. The function γ_{μ} is slowly varying and the function ω_{μ} is regularly varying of index

(2.6)  ℑ(ω_{μ}) = \frac{σ(μ)}{1 + σ(μ)} .

8. The function γ_{μ} is ultimately positive and infinite and we have

(2.7)  γ_{μ}(x) − μ(x) = O_{x→+∞} (1)

9. We have ultimately

(2.8)  ω′_{μ}(N_{μ}(x)) = \frac{e^{−γ_{μ} (x)}}{e} ,
(2.9)  γ_{μ}(x)) = ln(h_{μ}^{⟨−1⟩} (x)).

10. The following relations hold for every α ∈ ℝ_{+}^{∗}

(2.10)  μ(αx) − μ(x) = O_{x→+∞} (1),
(2.11)  lim_{x→+∞} \frac{e^{−αφ_{μ} (x)}}{φ′_{μ}(x)} = 0.

Proof. 1. Thanks to [4], the function h_{μ} is ultimately well defined by the system

x = exp[μ(t) + tμ′(t)], h_{μ}(x) = t^{2}μ′(t)

Consider then the function μ̄: x ↦ μ(x) + ln(x). It belongs to the Hardy field ℵ and we have σ(μ̄) = \frac{σ(μ)}{σ(μ)+1} ∈ ]0, +∞[ . It follows that the function hμ̄ is ultimately well defined by the system

x = exp[μ̄(t) + tμ̄′(t)], hμ̄(x) = t^{2} μ̄′(t)

that is by the system

x = et exp[μ(t) + tμ′(t)], hμ̄(x) = t + t^{2}μ′(t)

But we know that hμ̄ = ω_{μ}, thence the function ω_{μ} is ultimately well defined by the system

x = et exp[μ(t) + tμ′(t)], ω_{μ}(x) = t + t^{2}μ′(t)

the parameter t being ultimately uniquely determined by x.

On the other hand, since

t^{2}μ′(t) ∼_{t→+∞} \frac{1}{σ(μ)} t, exp[μ(t) + tμ′(t)] ∼_{t→+∞} e^{\frac{1}{σ(μ)}} e^{μ(t)}

it follows that lim_{x→+∞} h_{μ}(x) = +∞. Consequently h_{μ} is ultimately positive and infinite. Thence according to Theorem 2.1 above, the function h_{μ} has an inverse h_{μ}^{⟨−1⟩} which belongs to a Hardy field.

2. It follows from the definition of the function ω_{μ} that it is ultimately of class C^{1}. Direct computations from the system (2.2) prove then that the function ω′_{μ} has ultimately the following parametrical representation

(2.12)  x = et exp[μ(t) + tμ′(t)], ω′_{μ}(x) = \frac{1}{e exp[μ(t) + tμ′(t)]}

It follows that the function ω′_{μ} is ultimately strictly decreasing. Then that the function ω_{μ} is ultimately strictly concave.

3. Direct computations from system (2.2) lead to the system representing ultimately the function φ_{μ}.

4. It is clear that the function F_{1} : t → e^{μ(t)+tμ′(t)} which belongs as a germ at +∞ to the Hardy field ℵ, is ultimately strictly increasing and satisfies lim_{x→+∞} F_{1}(t) =

+∞. Thence, according to Theorem 2.1, the function F_{1} has an inverse g belonging to a Hardy field ℵ_{1} which contains the identity.

The function h_{μ} is ultimately of the class C^{1}, and, according to (2.5), we have ultimately

h′_{μ}(x) = \frac{\frac{d(t^{e} μ′(t))}{dt} (g(x))}{\frac{d(e^{μ(t)+tμ′(t)} )}{dt} (g(x))} = \frac{g(x)}{e^{μ(g(x))+g(x)μ′(t(x))}} = \frac{g(x)}{x}

Hence we have ultimately xh′_{μ}(x) = g(x). It follows, according to Theorem 2 above, that ℵ_{1}[h_{μ}] is an integral domain whose fraction field ℵ_{1}(h_{μ}) is a Hardy field which contains the function h_{μ} as a germ at +∞. By a similar proof we obtain that hμ̄ belongs to a Hardy field. Since ω_{μ} = hμ̄ it follows then that the function ω_{μ} belongs as a germ at +∞ to a Hardy field.

It is obvious that the function φ_{μ} belongs to the same Hardy field ℵ as ω_{μ}. Furthermore direct computations, based on the system representing φ_{μ} on some neighborhood of +∞, show that φ_{μ} is infinite and ultimately positive. It follows then that φ_{μ} is ultimately strictly increasing. Thence φ_{μ} is a diffeomorphism between neighborhoods of +∞ whose inverse N_{μ} belongs to a Hary field. It is clear that the function N_{μ} is ultimately well defined by the system

x = t^{2}μ′(t), N_{μ}(x) = et exp[μ(t) + tμ′(t)]

the parameter t being ultimately uniquely determined by x.

Direct computations from systems (2.1), (2.2), (2.4) prove that we have ultimately γ_{μ}(h_{μ}(x)) = ln(x), that is γ_{μ}(x) = ln(h_{μ}^{⟨−1⟩} (x))

It follows, according to Theorem 3 above, that there exists a Hardy field containing the function γ_{μ}. It follows also from the relation (2.9) that γ_{μ} is ultimately positive and infinite.

Direct computations from systems (2.1), (2.2), (2.4) prove also that relation (2.8) holds ultimately.

5. The function ω_{μ} belongs to a Hardy field and we have

lim_{x→+∞} \frac{xω′_{μ}(x)}{ω_{μ}(x)} = lim_{x→+∞} \frac{t_{1}(x)}{t_{1}(x) + t_{1}(x)^{2}μ′(t_{1}(x))}

= lim_{x→+∞} \frac{1}{1 + t_{1}(x)μ′(t_{1}(x))} = \frac{σ(μ)}{1 + σ(μ)}

Thence the function ω_{μ} is regularly varying with index ℑ(ω_{μ}) = \frac{σ(μ)}{1+σ(μ)} . 6. The function γ_{μ} belongs as a germ at +∞ to a Hardy field and we have

lim_{x→+∞} \frac{xγ′_{μ}(x)}{γ_{μ}(x)} = lim_{t→+∞} \frac{tμ′(t)}{μ(t) + tμ′(t)} = lim_{t→+∞} \frac{\frac{tμ′(t)}{μ(t)}}{1 + \frac{tμ′(t)}{μ(t)}} = 0

Thence γ_{μ} is slowly varying.

7. Since γ_{μ} is slowly varying it follows, according to Theorem 2.4 above, that we have ultimately 0 ⩽ γ_{μ}(x) ⩽ \sqrt{x}. It follows that γ_{μ}(x) = o_{x→+∞} (x).

On the other hand, according to (2.1), we have ultimately

γ_{μ}(x) − μ(x) = μ(t_{0}(x)) + t_{0}(x)μ′(t_{0}(x)) − μ(x)

= \frac{(t_{0}(x) − x)}{v} vμ′(v) + t_{0}(x)μ′(t_{0}(x))

where v lies between x and t_{0}(x). Since

x = t_{0}(x)^{2}μ′(t_{0}(x)) ∼_{x→+∞} \frac{1}{σ(μ)} t_{0}(x)

it follows that \frac{t_{0} (x)−x}{v} = O_{x→+∞} (1). Consequently we have

γ_{μ}(x) − μ(x) = O_{x→+∞} (1)

HOLOMORPHIC SERIES EXPANSION OF FUNCTIONS OF CARLEMAN TYPE 253

8. We have

lim_{x→+∞} \frac{e^{−αφ_{μ} (x)}}{φ′_{μ}(x)} = lim_{t→+∞} \frac{e[1 + 2tμ′(t) + t^{2}μ′′(t)] exp[μ(t) + tμ′(t) − αt^{2}μ′(t)]}{2tμ′(t) + t^{2}μ′′(t)}

Thence we have by virtue of L’Hopital’s rule that

lim_{t→+∞} −t^{2}μ′′(t) = lim_{t→+∞} tμ′(t) = lim_{t→+∞} \frac{μ(t)}{lnt} = \frac{1}{σ(μ)}

Consequently the following estimate holds

\frac{e[1 + 2tμ′(t) + t^{2}μ′′(t)] exp[μ(t) + tμ′(t) − αt^{2}μ′(t)]}{2tμ′(t) + t^{2}μ′′(t)}

∼_{t→+∞} e(1 + σ(μ))e^{\frac{1}{σ(μ)}} exp[μ(t) − αt^{2}μ′(t)].

But ℑ(μ) = 0 and ℑ(t ↦ αt^{2}μ′(t)) = 1, hence we have, according to Theorem 2.4 above, that μ(t) = O_{t→+∞} (αt^{2}μ′(t)). It follows that

lim_{t→+∞} e(1 + σ(μ))e^{\frac{1}{σ(μ)}} exp[μ(t) − αt^{2}μ′(t)] = 0.

Consequently we have

lim_{x→+∞} \frac{e^{−αφ_{μ} (x)}}{φ′_{μ}(x)} = 0.

On the other hand, according to the mean value theorem, we have for all x > 0

|μ(αx) − μ(x)| = |α − 1||xμ′(u)| = |α − 1|\frac{x}{u}|uμ′(u)|

where u lies between αx and x. It follows that

|μ(αx) − μ(x)| ⩽ |α − 1| max( α, 1/α) |uμ′(u)|

Since lim_{s→+∞} tμ′(t) = \frac{1}{σ(μ)} < +∞, it follows then that

μ(αx) − μ(x) = O_{x→+∞} (1)  □

Proposition 2.2. Let I be a nontrivial compact interval of ℝ and a ∈ I. The so-called Borel mapping T : C_{M_{μ}} (I) → Λ_{M_{μ}} , f ↦ (f^{(n)} (a))_{n∈ℕ} is surjective.

Proof. Following Petzsche [17, p. 300], we set

m_{p}^{∗} := \frac{M_{p}}{pM_{p−1}} , p ∈ ℕ^{∗}.

We have then for every p ∈ ℕ^{∗}

\frac{m_{2p}^{∗}}{m_{p}^{∗}} = \frac{1}{2} \frac{M_{2p}/M_{2p−1}}{M_{p}/M_{p−1}} = \frac{2^{2p}p^{2p}}{2(2p − 1)^{2p−1}} \frac{(p − 1)^{p−1}}{p^{p}}

× exp[2pμ(2p) − (2p−1)μ(2p−1) − pμ(p) − (p−1)μ(p−1)]

= \frac{(1 − \frac{1}{p})^{p−1}}{(1 − \frac{1}{2p})^{2p−1}} exp[2pμ(2p) − (2p−1)μ(2p−1) − pμ(p) − (p−1)μ(p−1)] .

But we have

2pμ(2p) − (2p − 1)μ(2p − 1) = z_{2p}μ′(z_{2p}) + μ(z_{2p}),
pμ(p) − (p − 1)μ(p − 1) = z_{p}μ′(z_{p}) + μ(z_{p})

where z_{2p} ∈ [2p−1, 2p] and z_{p} ∈ [p−1, p]. It follows that there exists w_{p} ∈ [p−1, 2p] such that

2pμ(2p) − (2p − 1)μ (2p − 1) − [pμ(p) − (p − 1)μ(p − 1)]

= z_{2p}μ′(z_{2p}) + μ(z_{2p}) − (z_{p}μ′(z_{p}) + μ(z_{p}))

= (z_{2p} − z_{p})(w_{p}μ′′(w_{p}) + 2μ′(w_{p}))

= (z_{2p} − z_{p})μ′(w_{p})[ \frac{w_{p}μ′′(w_{p}) + μ′(w_{p})}{μ′(w_{p})} + 1]

On the other hand, the limit lim_{x→+∞} \frac{xμ′′(x))+μ′(x)}{μ′(x)} exists and we have

lim_{x→+∞} \frac{xμ′(x)}{μ(x)} = 0.

From L’Hopital’s rule, it follows

(2.13)  lim_{x→+∞} \frac{xμ′′(x)) + μ′(x)}{μ′(x)} = 0

Furthermore we have z_{2p} − z_{p} ⩾ \frac{1}{2}w_{p} − 1. Thence we have for large values of p

(2.14)  (z_{2p} − z_{p})μ′(w_{p}) ⩾ \frac{1}{2}[ \frac{w_{p} − 2}{w_{p}} ]w_{p}μ′(w_{p})

We conclude from (2.13) and (2.14) that

liminf_{p→+∞}[2pμ(2p) − (2p − 1)μ(2p − 1)] − [pμ(p) − (p − 1)μ(p − 1)] ⩾ \frac{1}{2σ(μ)}

It follows that

liminf_{p→+∞} \frac{m_{2p}^{∗}}{m_{p}^{∗}} ⩾ e^{\frac{1}{2σ(μ)}} > 1.

Thence a slight refinement of a theorem in [17, pp. 300 and 311] yields that the Borel mapping T is surjective. □

Direct computations show that μ and γ_{μ} can be extended to ℝ_{+}^{∗} in a way to be functions of class C^{1} on ℝ_{+}^{∗} such that −ε ⩽ μ(x) − γ_{μ}(x) ⩽ ε,x ∈ ℝ_{+}^{∗} where ε is a positive constant. From now on we will do so and we will set for every A > 0, n ∈ ℕ and for every nonempty subset S of ℂ

S_{μ,A,n} := S_{Ae^{−μ(n)}} , S_{γ_{μ} ,A,n} := S_{Ae^{−γμ (n)}} .

Thence the following inclusions hold for all n ∈ ℕ.

S_{γ_{μ} ,Ae^{−ε} ,n} ⊂ S_{μ,A,n} ⊂ S_{γ_{μ} ,Ae^{ε},n} .

3. Statement of the main result

The main result of this paper is the following.

Theorem 3.1. 1. Let f ∈ C_{M_{μ}} ([−1, 1]); then there exists constants C > 0, A > 0, 0 < ρ < 1 and a sequence (P_{n})_{n⩾1} of rational functions defined on ℂ∖{i, −i} such that ∑ P_{n} is uniformly convergent on [−1, 1] to f and

∥P_{n}∥_{∞,[−1,1]_{μ,A,n}} ⩽ Cρ^{n}, n ∈ ℕ, f(x) = ∑_{n=1}^{∞} P_{n}(x), x ∈ [−1, 1]

2. Conversely, let us assume that there exist some constants C > 0, A > 0, 0 < ρ < 1 and a sequence f_{n} ∈ O([−1, 1]_{μ,A,n} ) of holomorphic functions such that ∥f_{n}∥_{∞,[−1,1]_{μ,A,n}} ⩽ Cρ^{n}, n ∈ ℕ^{∗}. Then the function series ∑ f_{n} is uniformly convergent on [−1, 1] to a function f which belongs to the Carleman class C_{M_{μ}} ([−1, 1]).

4. Proof of the main result

4.1. Direct part.

Proposition 4.1. Let g: [−π, π] −→ C be a restriction of a 2π-periodic function of class C^{∞} on ℝ. Let us assume that g ∈ C_{M_{μ}} ([−π, π]); then there exist constants A > 0, C > 0, 0 < ρ < 1 and a sequence (g_{n})_{n⩾0} of rational functions defined on ℂ^{∗} such that

∥g_{n}∥_{∞,K_{γμ ,A,n}} ⩽ Cρ^{n}, n ∈ ℕ, g(θ) = ∑_{n=0}^{∞} g_{n}(e^{iθ}), θ ∈ [−π, π]

where K_{γ_{μ} ,A,n} := {z ∈ ℂ, 1 − Ae^{−γ_{μ} (n)} < |z| < 1 + Ae^{−γ_{μ} (n)} }.

Proof. The Fourier series expansion of g can be written for all θ ∈ [−π, π] as

g(θ) = ∑_{p∈ℤ} a_{p}e^{ipθ} where a_{p} = \frac{1}{2π} ∫_{−π}^{π} g(θ)e^{−ipθ} dθ, p ∈ ℤ.

According to [16], the following estimations hold

(4.1)  |a_{p}| ⩽ C_{0}e^{−C_{1} ω_{μ} (|p|)} , p ∈ ℤ

with some constants C_{0}, C_{1} > 0.

Let us set for all z ∈ ℂ^{∗} and n ∈ ℕ^{∗}

g_{0}(z) := ∑_{|p|<N_{μ} (1)} a_{p}z^{p}, g_{n}(z) := ∑_{N_{μ} (n)⩽|p|<N_{μ} (n+1)} a_{p}z^{p}.

Then for all n ∈ ℕ^{∗}, g_{n} is a rational function defined on ℂ^{∗}. Furthermore the following estimates hold

(4.2)  |g_{n}(z)| ⩽ C_{0} ∑_{N_{μ} (n)⩽|p|<N_{μ} (n+1)} C_{0}e^{−C_{1} ω_{μ} (p)} (|z|^{p} + |z|^{−p} ), z ∈ ℂ^{∗}.

If z ∈ K_{\frac{C_{1}}{2e} ,n}, then the estimates become

|g_{n}(z)| ⩽ C_{0} ∑_{N_{μ} (n)⩽|p|<N_{μ} (n+1)} e^{−C_{1} ω_{μ} (p)} [( 1+ \frac{C_{1}}{2e} e^{−γ_{μ} (n)} )^{p} +( 1− \frac{C_{1}}{2e} e^{−γ_{μ} (n)} )^{−p} ].

We have for large values of n

( 1 − \frac{C_{1}}{2e} e^{−γ_{μ} (n)} )^{−1} ⩽ 1 + \frac{C_{1}}{e} e^{−γ_{μ} (n)} .

It follows that we have for such values of n

∥g_{n}∥_{∞K_{γμ , \frac{C_{1}}{2e} ,n}} ⩽ C_{0}(1+N_{μ}(n+1)−N_{μ}(n)) max_{N_{n} ⩽p<N_{n+1}} 2 exp[−C_{1}ω_{μ}(p)+C_{1}p\frac{e^{−γ_{μ} (n)}}{e} ].

On the other hand we have for n sufficiently large

\frac{e^{−γ_{μ} (n)}}{e} = ω′_{μ}(N_{μ}(n))

Consequently we have for such values of n

∥g_{n}∥_{∞,K_{γμ , \frac{C_{1}}{2e} ,n}} ⩽ C_{0}(1 + N_{μ}(n + 1) − N_{μ}(n))

max_{N_{μ} (n)⩽p<N_{μ} (n+1)} 2 exp[ − C_{1}(ω_{μ}(p)− ω′_{μ}(N_{μ}(n))p)]

But by virtue of Proposition 2.1, ω_{μ} is ultimately strictly concave. It follows that the function h_{n} : ℝ_{+}^{∗} → ℝ, x ↦ −C_{1}[ω(x) − ω′_{μ}(N_{μ}(n))x] is ultimately strictly concave, thence we have for large values of n that for all x ∈ [N_{μ}(n), N_{μ}(n + 1)] we have

h′_{n}(x) = −C_{1}[ω′(x) − ω′_{μ}(N_{μ}(n))] < 0

Thence the function h_{n} is, for large values of n, strictly decreasing on the interval [N_{μ}(n), N_{μ}(n+1)]. It follows that the following estimates hold for large values of n

∥g_{n}∥_{∞,K_{γμ , \frac{C_{1}}{2e} ,n}}
⩽ C_{0}[1 + N_{μ}(n + 1) − N_{μ}(n)] exp[−C_{1}(ω(N_{μ}(n)) − N_{μ}(n)ω′(N_{μ}(n)))]
⩽ C_{0}[1 + N_{μ}(n + 1) − N_{μ}(n)] exp[−C_{1}φ_{μ}(N_{μ}(n))]
⩽ C_{0}[1 + N_{μ}(n + 1) − N_{μ}(n)]e^{−C_{1} n}
⩽ C_{0}[e^{\frac{C_{1}}{2}} (N_{μ}(n + 1) − N_{μ}(n))e^{− \frac{C_{1}}{2} (n+1)} + 1]e^{− \frac{C_{1}}{2} n}

Since N_{μ} is ultimately strictly convex, we can write for large values of n

∥g_{n}∥_{∞,K_{γμ , \frac{C_{1}}{2e} ,n}} ⩽ C_{0}[e^{\frac{C_{1}}{2}} N′_{μ}(n + 1)e^{− \frac{C_{1}}{2} (n+1)} + 1]e^{− \frac{C_{1}}{2} n}

⩽ C_{0}[e^{\frac{C_{1}}{2}} \frac{e^{− \frac{C_{1}}{2} φ_{μ} (N_{μ} (n+1))}}{φ′_{μ}(N_{μ}(n + 1))} + 1]e^{− \frac{C_{1}}{2} n}

According to (2.11) we have

C_{0}[e^{\frac{C_{1}}{2}} \frac{e^{− \frac{C_{1}}{2} φ_{μ} (N_{μ} (n+1))}}{φ′_{μ}(N_{μ}(n + 1))} + 1]e^{− \frac{C_{1}}{2} n} ∼_{n→+∞} C_{0}e^{− \frac{C_{1}}{2} n}

Thence we have

∥g_{n}∥_{∞,K_{γμ , \frac{C_{1}}{2e} ,n}} ⩽ C_{2}e^{− \frac{C_{1}}{2} n} , n ∈ ℕ

where C_{2} > 0 is a constant. □

Proposition 4.2. Let f ∈ C_{M_{μ}} ([−1, 1]); then there exists a function F ∈ C_{M_{μ}} (ℝ) with support contained in the interval [−2, 2] and whose restriction to [−1, 1] is the function f.

Proof. According to Proposition 2.2, there exist F_{1} ∈ C_{M_{μ}} ([−3, −1] and F_{2} ∈ C_{M_{μ}} ([1, 3]) such that F_{1}^{(n)} (−1) = f^{(n)} (−1), F_{2}^{(n)} (1) = f^{(n)} (1), n ∈ ℕ. On the other hand, according to [22], there exists Φ ∈ C_{M_{μ}} (ℝ) with support contained in [−2, 2] such that Φ(x) = 1, x ∈ [−1, 1]. The function F defined by

\begin{aligned}F (x) & = f(x), & x ∈ [−1, 1] \\ F (x) & = F_{1}(x)Φ(x), & x ∈ [−3, −1] \\ F (x) & = F_{2}(x)Φ(x), & x ∈ [1, 3] \\ F (x) & = 0, & x ∈ ℝ ∖ [−3, 3] \end{aligned}

satisfies the required conditions. □

End of the proof of the direct part of the main theorem. Let f ∈ C_{M_{μ}} ([−1, 1]). There exists, according to Proposition 4.2, a function F ∈ C_{M_{μ}} (ℝ) whose support is contained in the interval [−2, 2] and whose restriction to [−1, 1] is the function f.

Let us consider the function g defined on the interval [−π, π] by

g(θ) = F (tan(θ/2)), θ ∈ ]−2 arctan(2), 2 arctan(2)[
g(θ) = 0,  θ ∈ ℝ ∖]−2 arctan(2), 2 arctan(2)[

According to Cartan [11, Theorem III, pp. 24–27], the restriction of g to the interval J := [−2 arctan(2), 2 arctan(2)] belongs to the Carleman class C_{M_{μ}} (J). But g is itself the restriction to [−π, π] of a 2π-periodic, of class C^{∞} which is vanishing on the set [−π, π] ∖ J. Thence g ∈ C_{M_{μ}} ([−π, π]).

According to Proposition 4.1 there exists constants 0 < A < 1, C > 0, 0 < ρ < 1 and a sequence (g_{n})_{n⩾1} of rational functions defined on ℂ^{∗} such that

∥g_{n}∥_{∞,K_{γμ ,A,n}} ⩽ Cρ^{n}, n ∈ ℕ, g(θ) = ∑_{n=0}^{∞} g_{n}(e^{iθ}), θ ∈ [−π, π].

Let x ∈ [−2, 2]. There exists a unique θ ∈ [−2 arctan(2), 2 arctan(2)] such that x = tan( \frac{θ}{2}), thence we have F (x) = g(θ) = ∑ _{n=1}^{+∞} g_{n}( \frac{i−x}{i+x} ). On the other hand let z ∈ ℂ be such that | Im(z)| < 1 (then z ∈ ℂ ∖ \{i, −i\}). Let us set ζ = \frac{i−z}{i+z} ; then we have | Im(z)| ⩾ \frac{|1−|ζ||}{1+|ζ|} . It follows that the following implication holds for every A′ ∈ ]0, 1[

| Im(z)| ⩽ A′e^{−γ_{μ} (n)} ⇒ \frac{1 − A′e^{−γ_{μ} (n)}}{1 + A′e^{−γ_{μ} (n)}} ⩽ |ζ| ⩽ \frac{1 + A′e^{−γ_{μ} (n)}}{1 − A′e^{−γ_{μ} (n)}}

If we choose A′ ∈ ]0, 1[ sufficiently small, then we will obtain for every n ∈ ℕ

0 < 1 − Ae^{−γ_{μ} (n)} < \frac{1 − A′e^{−γ_{μ} (n)}}{1 + A′e^{−γ_{μ} (n)}} ⩽ \frac{1 + A′e^{−γ_{μ} (n)}}{1 − A′e^{−γ_{μ} (n)}} < 1 + Ae^{−γ_{μ} (n)}

Let us set B_{n} := {z ∈ ℂ : | Im(z)| < A′e^{−γ_{μ} (n)} }. Thence the points i and −i belong to ℂ ∖ B_{n} and we have \frac{i−z}{i+z} ∈ K_{γ_{n} ,A,n} , z ∈ B_{n}. For each n ∈ ℕ, the function P_{n} defined on ℂ\\{i, −i\} by P_{n}(z) = g_{n}(\frac{i−z}{i+z} ) is a rational function satisfying

∥P_{n}∥_{∞,B_{n}} ⩽ Cρ^{n}, n ∈ ℕ

We have also for all x ∈ [−2, 2] that F (x) = ∑ _{n=1}^{∞} P_{n}(x). But [−1, 1]_{γ_{μ} ,A′,n} ⊂ B_{n} for all n ∈ ℕ; thence we have

f(x) = ∑_{n=1}^{∞} P_{n}(x), x ∈ [−1, 1], ∥P_{n}∥_{∞,[−1,1]_{γμ ,A′,n}} ⩽ Cρ^{n}, n ∈ ℕ.

Then, it follows

f(x) = ∑_{n=1}^{∞} P_{n}(x), x ∈ [−1, 1],  ∥P_{n}∥_{∞,[−1,1]_{μ,Ae^{−ε′} ,n}} ⩽ Cρ^{n}, n ∈ ℕ.  □

4.2. Converse part.

Proof. Let A > 0 and for each n ∈ ℕ, a function f_{n} : [−1, 1]_{μ,A,n} → ℂ which is holomorphic on [−1, 1]_{μ,A,n} such that

f_{n} ∈ O([−1, 1]_{μ,A,n} ), n ∈ ℕ^{∗}, ∥f_{n}∥_{∞,[−1,1]_{μ,A,n}} ⩽ Cρ^{n}, n ∈ ℕ^{∗}.

It follows that ∥f_{n}∥_{∞,[−1,1]_{γμ ,Ae^{−ε} ,n}} ⩽ Cρ^{n}, n ∈ ℕ^{∗}. Thence the function series ∑ f_{n} _{|[−1,1]} converges uniformly on [−1, 1] to a continuous function f.

We have [−1, 1] ⊂ [−1, 1]_{γ_{μ} , \frac{A}{2} e^{−ε} ,n} ⊂ [−1, 1]_{γ_{μ} ,Ae^{−ε} ,n} . Cauchy’s inequalities allow us to write for all p ∈ ℕ

(4.3)  ∥f_{n}^{(p)} ∥_{∞,[−1,1]} ⩽ Cp!( \frac{2}{A}e^{ε})^{p} exp[pγ_{μ}(n) − ln(ρ^{−1/2} )n]ρ^{−n/2} .

On the other hand the supremum, for sufficiently large p ∈ ℕ, of the function u ↦ pγ_{μ}(u) − ln(1/\sqrt{ρ})u on [0, +∞] is reached in the real u_{p} > 0 that satisfies γ′_{μ}(u_{p}) = \frac{1}{p} ln(1/\sqrt{ρ}). Since for sufficiently large p ∈ ℕ, we have γ′_{μ}(u_{p}) = 1/t_{0}(u_{p}), it follows that t_{0}(u_{p}) = p/ ln(1/\sqrt{ρ}). Consequently we can write

(4.4)  sup_{n∈ℕ}[pγ_{μ}(n) − ln(1/\sqrt{ρ})n] ⩽ p(γ_{μ}(u_{p}) − u_{p}γ′_{μ}(u_{p}))
⩽ pμ(t_{0}(u_{p})) ⩽ pμ(p/ ln(1/\sqrt{ρ}))

Thence we have for p ∈ ℕ sufficiently large we have for all n ∈ ℕ

∥f_{n}^{(p)}∥_{∞,[−1,1]} ⩽ Cp!( \frac{2}{A}e^{ε})^{p}\sqrt{ρ}^{n}e^{pμ(p/ ln(1/\sqrt{ρ}))}

It follows that the function series ∑ f_{n}^{(p)} are for sufficiently large values of p normally convergent. Thence the function f is of class C^{∞} on [−1, 1] and we have

∥f^{(p)}∥_{∞,[−1,1]} ⩽ \frac{2C}{A(1 − \sqrt{ρ})} ( \frac{2}{A})^{p}p! exp[p(μ(p/ ln(1/\sqrt{ρ})) − μ(p))]e^{pμ(p)}
⩽ B^{p+1} p^{p}e^{pμ(p)}

for some constant B > 0. Thence we have f ∈ C_{M_{μ}} ([−1, 1]). □

5. Application: Alternative construction of Dyn’kin’s pseudoanalytic extension for the Carleman class C_{M_{μ}} ([−1, 1])

Corollary 5.1. Let be f ∈ C_{M_{μ}} ([−1, 1]). There exists a function F ∈ C^{∞} (ℂ) with compact support such that

F |_{[−1,1]} = f, |¯∂F (z)| ⩽ C_{1}H_{μ}[ \frac{C_{2}}{ρ(z, [−1, 1])} ], z ∈ ℂ ∖ [−1, 1]

where C_{1}, C_{2} > 0 are constants.

Proof. According to the main result there exist constants A ∈ ]0, 1[, C > 0, ρ ∈ ]0, 1[, and a sequence of rational functions (f_{n})_{n∈ℕ} defined on some strip B := {z ∈ ℂ : | Im(z)| ⩽ A} such that

∥f_{n}∥_{∞,[−1,1]_{μ,A,n}} ⩽ Cρ^{n}, n ∈ ℕ^{∗},  ∑_{n=1}^{+∞} f_{n|[−1,1]} = f.

It follows that ∥f_{n}∥_{∞,[−1,1]_{γμ ,Ae^{−ε} ,n}} ⩽ Cρ^{n}, n ∈ ℕ^{∗}.

On the other hand, there exists, for each n ∈ ℕ^{∗}, a function θ_{n} : ℂ → [0, 1] of class C^{∞} on ℂ (ℂ is here identified with ℝ^{2}) and a family of positive constants (L_{α})_{α∈ℕ^{2}} [22] such that

θ_{n}(z) = 1, z ∈ [−1, 1]_{μ, \frac{A}{8} ,n}
θ_{n}(z) = 0, z ∈ ℂ ∖ [−1, 1]_{μ, \frac{A}{2} ,n}
|D^{α} θ_{n}(z)| ⩽ L_{α} e^{|α|μ(n)} , α ∈ ℕ^{2}, z ∈ ℝ^{2}

where |α| := p + q and D^{α} := \frac{∂^{p+q}}{∂x^{p} ∂y^{q}} for α = (p, q).

We denote by F_{n} the function defined by

F_{n}(z) = θ_{n}(z)f_{n}(z), z ∈ [−1, 1]_{γ_{μ} ,A,n}
F_{n}(z) = 0, z ∈ ℂ ∖ [−1, 1]_{γ_{μ} ,A,n}

The function F_{n} is of class C^{∞} on ℂ and satisfies the condition

F_{n}|_{[−1,1]_{μ, \frac{A}{8} ,n}} = f_{n} |_{[−1,1]_{μ, \frac{A}{8} ,n}} .

Since ∥F_{n}∥_{∞,ℂ} ⩽ Cρ^{n}, n ∈ ℕ, it follows that the function series ∑ F_{n} is uniformly convergent on ℂ to a continuous function F with compact support contained in [−1, 1]_{A}. Furthermore it is clear that F is an extension to ℂ of f.

Let α ∈ ℕ^{2}, n ∈ ℕ and z ∈ ℂ. If z ∈ ℂ ∖ [−1, 1]_{μ, \frac{A}{2} ,n} , then we have D^{α} F_{n}(z) = 0 . But when z ∈ [−1, 1]_{μ, \frac{A}{8} ,n} we can write, in view of Cauchy’s inequalities and inequality (4.4)

|D^{α} F_{n}(z)| ⩽ ∑_{β⩽α} (_{β}^{α})|D^{β}θ_{n}(z)||D^{α−β} f_{n}(z)|

⩽ ∑_{β⩽α} (_{β}^{α})L_{β}e^{|β|μ(n)} |D^{α−β} f_{n}(z)|

⩽ ∑_{β⩽α} (_{β}^{α})L_{β}e^{|β|ε} e^{|β|γ_{μ} (n)} |f_{n}^{(|α|−|β|)} (z)|

⩽ ∑_{β⩽α} (_{β}^{α})L_{β}e^{|β|ε} e^{|β|γ_{μ} (n)} C(4/A)^{|α|−|β|}

· (|α| − |β|)!\sqrt{ρ}^{n} exp[(|α| − |β|)γ_{μ}(n) − ln(1\sqrt{ρ})n]

⩽ ∑_{β⩽α} (_{β}^{α})L_{β}e^{|β|ε} e^{|β|γ_{μ} (n)} C(4/A)^{|α|−|β|}

· (|α| − |β|)!\sqrt{ρ}^{n} exp[ sup_{m∈ℕ} ((|α| − |β|)γ_{μ}(m) − ln(1/\sqrt{ρ})m)]

⩽ \sqrt{ρ}^{n} ∑_{β⩽α} C(_{β}^{α})e^{|β|ε} L_{β}(|α| − |β|)!(4/A)^{|α|−|β|}

· exp[(|α| − |β|)μ( \frac{(|α| − |β|)}{ln(1/\sqrt{ρ})} ) ]

It follows that the function series ∑ D^{α} F_{n}(z) is for all α ∈ ℕ^{2} normally convergent on ℂ. Thence the function F = ∑ _{n=1}^{+∞} F_{n} is of class C^{∞} on ℂ.

Let z ∈ ℂ ∖ [−1, 1]. Then we have ¯∂F (z) = ∑ _{n=1}^{+∞} ¯∂F_{n}(z). On the other hand, we have

¯∂F_{n}(z) = 0 if ρ(z, [−1, 1]) ∈ [0, \frac{A}{8}e^{−ε} e^{−γ_{μ} (n)} [ ∪ ]Ae^{−ε} e^{−γ_{μ} (n)} , +∞[.

If ρ(z, [−1, 1]) ∈ [ \frac{A}{8}e^{−μ(n)} , Ae^{−μ(n)} [, then, again by virtue of (4.4), we have

|¯∂F_{n}(z)| = |f_{n}(z)||¯∂θ_{n}(z)|

⩽ \frac{C}{2}ρ^{n}( | \frac{∂θ_{n}}{∂x} (z)| + | \frac{∂θ_{n}}{∂y} (z)|)

⩽ \frac{C}{2}(L_{(1,0)} + L_{(0,1)} )e^{ε}e^{γ_{μ} (n)− \frac{1}{2} ln( \frac{1}{ρ} )n} \sqrt{ρ}^{n}

⩽ \frac{C}{2}(L_{(1,0)} + L_{(0,1)} )e^{ε}e^{μ(2/ ln(1/\sqrt{ρ}))} \sqrt{ρ}^{n}

Let us set

A_{1} := \frac{C}{2}(L_{(1,0)} + L_{(0,1)} )e^{ε}e^{μ(2/ ln(1/\sqrt{ρ}))} , λ := − ln \sqrt{ρ} > 0

Thence the following estimate holds

|¯∂F (z)| ⩽ ∑_{\frac{A}{8} e^{−μ(n)} ⩽ρ(z,[−1,1])⩽Ae^{−μ(n)}} A_{1}e^{−λn}

⩽ A_{1} ∑_{\frac{A}{8ρ(z,[−1,1])} ⩽e^{μ(n)}} e^{−λn}

⩽ A_{1} ∑_{\frac{A}{8e^{ε} ρ(z,[−1,1])} ⩽e^{γμ (n)}} e^{−λn}

It follows that if z is sufficiently close to [−1, 1], then the last estimate will become

|¯∂F (z)| ⩽ A_{1} ∑_{h_{μ} ( \frac{A}{8e^{ε} ρ(z,[−1,1])} )⩽n} e^{−λn}

⩽ \frac{A_{1}}{1 − e^{−λ}} exp[ − λh_{μ}( \frac{A}{8e^{ε}ρ(z, [−1, 1])} ) ]

But we know that the function h_{μ} is regularly varying. Thence there exists a constant A_{2} > 0 such that we have ultimately

λh_{μ}( \frac{A}{8e^{ε}} x) ⩾ h_{μ}(A_{2}x)

Consequently we have for z sufficiently close to [−1, 1]

|¯∂F (z)| ⩽ \frac{A_{1}}{1 − e^{− \frac{λ}{2}}} exp[ − h_{μ}( \frac{A_{2}}{ρ(z, [−1, 1])} ) ]

Thence there exists a constant A_{3} > 0 such that

|¯∂F (z)| ⩽ A_{3}H_{μ}( \frac{A_{2}}{ρ(z, [−1, 1])} ) , z ∈ ℂ

The proof of the corollary is complete. □

Acknowledgment. I would like to express my profound gratitude to Professor Jean Pierre Demailly, to Professor Alexander Borichev and to Professor Konstantin Mikhailovich Dyakonov for their precious assistance and their valuable helps without which my whole mathematical works would not have been realized.

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Ibn Tofail University (Received 10 12 2015) Departement of Mathematics Faculty of Sciences Kenitra Morocco

    hzoubeir2014@gmail.com