PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 299–304 DOI: 10.2298/PIM1614299Z

ABOUT A CONJECTURE ON DIFFERENCE EQUATIONS IN QUASIANALYTIC CARLEMAN CLASSES

Hicham Zoubeir

Abstract. We consider the difference equation ∑ _{j=1}^{q} a_{j} (x)φ(x + α_{j} ) = χ(x) where α_{1} < ⋯< α_{q} (q ⩾ 3) are given real constants, a_{j} (j = 1, …, q) are given holomorphic functions on a strip ℝ_{δ} (δ > 0) such that a_{1} and a_{q} vanish nowhere on it, and χ is a function belonging to a quasianalytic Carleman class C_{M} {ℝ}. We prove, under a growth condition on the functions a_{j} , that the difference equation above is solvable in C_{M} {ℝ}.

1. Introduction

Belitskii, Dyn’kin and Tkachenko in [1] formulated the following conjecture.

Conjecture. Let χ, a_{j}, j = 1, …, q, be functions in a Carleman class C_{M} {ℝ} such that a_{1} and a_{q} nowhere vanish on ℝ, and α_{1} < ⋯< α_{q} some real numbers. Then the difference equation

(1.1)  ∑_{j=1}^{q} a_{j} (x)φ(x + α_{j}) = χ(x)

is solvable in the Carleman class C_{M} {ℝ}.

In that paper, the authors, relying on a result of decomposition in Carleman classes, proved the conjecture in the particular cases where the coefficients a_{j} are constants or when the coefficients are variables with q = 2. They also suggested that the same method could be used to show the solvability of equation (1.1) in a quasianalytic Carleman class C_{M} \{ℝ\}, if we assume that the functions \frac{1}{a_{1}} , \frac{1}{a_{q}} , \frac{a_{2}}{a_{1}} , …, \frac{a_{q}}{a_{1}} , \frac{a_{1}}{a_{q}} , …, \frac{a_{q−1}}{a_{q}} (q ⩾ 3) can be continued in a strip ℝ_{δ} := \{z ∈ ℂ : | Im(z)| < δ} as analytic functions increasing on ℝ_{δ}, not too rapidly in infinity. As an example of such coefficients, they mentioned the class of rational functions. Our aim here is to give a precise meaning to this assertion, by proving that the result is true even if the functions \frac{1}{a_{1}} , \frac{1}{a_{q}} , \frac{a_{2}}{a_{1}} , …, \frac{a_{q}}{a_{1}} , \frac{a_{1}}{a_{q}} , …, \frac{a_{q−1}}{a_{q}} have more rapid increase in infinity, provided that it is of the form exp(e^{C| Re(z)|} ) where C > 0 is a constant.

2010 Mathematics Subject Classification: Primary 30H05; Secondary 30B10, 30D05.

Key words and phrases: difference equations, quasianalytic Carleman classes.

Communicated by Stevan Pilipović.

2. Notations, definitions and statement of the main result

We set for every ρ > 0, a ⩾ 0

ℝ_{ρ} := {z ∈ ℂ : | Im(z)| < ρ}, ℝ_{ρ}^{±} := {z ∈ ℝ_{ρ} : ± Re(z) > ρ}
ℝ_{ρ,a} := {z ∈ ℝ_{ρ} : | Re(z)| ⩽ a}
Δ_{ρ} := {z ∈ ℂ : |z| < ρ}, Δ_{ρ}^{±} := {z ∈ Δ_{ρ} : ± Re(z) ⩽ 0}
Γ_{ρ} := {z ∈ ℂ : |z| = ρ}, Γ_{ρ}^{±} := {z ∈ Γ_{ρ} : ± Re(z) ⩽ 0}

For every nonempty subset V of ℂ and every z ∈ ℂ and n ∈ ℕ^{∗} we set

V^{(0)} := V, V^{(n)} := {u_{1} + ⋯+ u_{n} : u_{j} ∈ V, j = 1, …, n}, n ⩾ 1
z + V := {z + u : u ∈ V }, z − V := {z − u : u ∈ V }

Denote by dm(ζ) the Lebesgue measure on ℂ. Let S be a nonempty subset of ℂ. By O(S) we denote the set of holomorphic functions on some neighborhood of S. Let F : U ⊂ ℂ → ℂ be a function of class C^{1} on an open subset U of ℂ. For all z ∈ U we set

\overline{∂}F (z) := \frac{1}{2}[ \frac{∂F}{∂x} (z) + i \frac{∂F}{∂y} (z)];

\overline{∂} is called the operator of Cauchy–Riemann.

Let M := (M_{n})_{n⩾0} be a sequence of strictly positive real numbers. The Carleman class C_{M} {ℝ} is the set of all functions f : ℝ → ℂ of class C^{∞} such that

∥f^{(n)} ∥_{∞,I} ⩽ C_{I} ρ_{I}^{n}M_{n}, n ∈ ℕ

for every compact interval I of ℝ with some constants C_{I} , ρ_{I} > 0. The Carleman class C_{M} {ℝ} is said to be quasinalytic if every function f ∈ C_{M} {ℝ} such that f^{(n)} (u) = 0 for some u ∈ ℝ and every n ∈ ℕ is identically equal to 0. The Carleman class C_{M} {ℝ} is called regular when the following conditions hold

( \frac{M_{n+1}}{(n + 1)!} )^{2} ⩽ \frac{M_{n}}{n!} \frac{M_{n+2}}{(n + 2)!} , n ∈ ℕ
sup_{n∈ℕ}( \frac{M_{n+1}}{(n + 1)M_{n}} ) ^{\frac{1}{n}} < +∞,
lim_{n→+∞} Mn^{\frac{1}{n}} = +∞

To the Carleman C_{M} {ℝ} we associate its weight H_{M} defined by the following relation

H_{M} (x) := lim_{n∈ℕ}[ \frac{M_{n}}{n!} x^{n}], x > 0

In this paper, the following result will play a crucial role.

Theorem 2.1. [3] We assume that the Carleman class C_{M} {ℝ} is regular. A function f : ℝ → ℂ belongs to C_{M} {ℝ} if and only if there exists for every compact interval I of ℝ a compactly supported function F : ℂ → ℂ of class C^{1} such that F is an extension to ℂ of the restriction to I of the function f and satisfies the following estimate

|\overline{∂}F (z)| ⩽ A_{I} H_{M} (B_{I} | Im(z)|), z ∈ ℂ

where A_{I} , B_{I} > 0 are constants.

Throughout the paper, we assume that the Carleman class C_{M} {ℝ} is regular and quasianalytic. Our main result is the following.

Theorem 2.2. Let q ∈ ℕ^{∗} ∖ {1, 2}, δ > 0, χ ∈ C_{M} {ℝ}, and a_{j} ∈ O(ℝ_{δ}) (j = 1, …, q) such that a_{1} and a_{q} nowhere vanish on ℝ_{δ}. We assume that the following growth condition holds

(2.1)  sup_{z∈ℝ_{δ}}( ∑_{j=2}^{q} | \frac{a_{j} (z)}{a_{1}(z)} | + ∑_{j=1}^{q−1} | \frac{a_{j} (z)}{a_{q}(z)} | + \frac{1}{|a_{1}(z)|} + \frac{1}{|a_{q}(z)|}) e^{−e^{C| Re z|}} < +∞

for a constant C > 0. Then difference equation (1.1) is solvable in the class C_{M} {ℝ}.

3. Proof of the main result

Let us first prove the following lemma.

Lemma 3.1. Given f ∈ C_{M} \{ℝ\}, C_{0} > 0 and ρ ∈ ]0; \frac{π}{2C_{0}} [, there exist two functions f_{±} : (ℂ ∖ Δ_{ρ}^{±} ) ∪ ℝ → ℂ which are holomorphic on ℂ ∖ (Γ_{ρ}^{±} ∪ Δ_{ρ}^{±} ) , whose restrictions to ℝ belong to C_{M} {ℝ}, and such that the following conditions hold

f(x) = f_{+} (x) + f_{−} (x), x ∈ [−ρ, ρ]
|f_{±} (z)| ⩽ D_{0} exp(− cos (ρC_{0})e^{C_{0} | Re(z)|} ), z ∈ ℝ_{ρ}^{±}

where D_{0} > 0 is a constant.

Proof. Since f belongs to C_{M} {ℝ}, there exists, according to Dyn’kin’s theorem [3], a compactly supported function F : ℂ → ℂ of class C^{1} such that F is an extension of the restriction of f to the interval [−ρ, ρ] and satisfies the following estimate

|\overline{∂}F (z)| ⩽ AH_{M} (B| Im(z)|), z ∈ ℂ

where A, B > 0 are constants. Following the same approach as that of [1, pp. 34,35], [2, pp. 148–150], but using the Cauchy–Pompeiu formula on the disk Δ_{ρ}, for the function exp(e^{C_{0} z} + e^{−C_{0} z} )f(z), we show that the functions

f_{±} (z) = \frac{1}{2iπ} exp(−e^{C_{0} z} − e^{−C_{0} z} ) ∫_{Γ_{ρ}^{±}} \frac{exp(e^{C_{0} ζ} + e^{−C_{0} ζ} )F (ζ)}{ζ − z} dζ

− \frac{1}{π} exp(−e^{C_{0} z} − e^{−C_{0} z} ) ∫∫_{Δ_{ρ}^{±}} \frac{exp(e^{C_{0} ζ} + e^{−C_{0} ζ} )\overline{∂}F (ζ)}{ζ − z} dm(ζ)

satisfy the required conditions. □

302 ZOUBEIR

Now we set

β_{j} := α_{j} − α_{1}, j = 2, …, q,  b_{j} (z):= −\frac{a_{j} (z)}{a_{1}(z)}, z ∈ ℝ_{δ}, j = 2, …, q
γ_{j} := α_{q} − α_{j} , j = 1, …, q − 1,  c_{j} (z) := −\frac{a_{j} (z)}{a_{q}(z)}, z ∈ ℝ_{δ}, j = 1, …, q − 1

Let C_{1} > C( \frac{β_{q}}{β_{2}} + \frac{γ_{1}}{γ_{q−1}} ) and δ_{0} ∈ ]0, min(δ, \frac{π}{2C_{1}} )[. Then according to the lemma above, there exists a constant D_{1} > 0 and two functions χ_{±} : (ℂ ∖ Δ_{δ_{0}}^{±}) ∪ ℝ → ℂ which are holomorphic on ℂ∖(Γ_{δ_{0}}^{±} ∪Δ_{δ_{0}}^{±}), whose restrictions to ℝ belong to C_{M} {ℝ}, and such that the following conditions hold (3.1)

χ(x) = χ_{+} (x) + χ_{−} (x), x ∈ [−δ_{0}, δ_{0}],
|χ_{±} (z)| ⩽ D_{1} exp(− cos(C_{1}δ_{0})e^{C_{1} | Re(z)|} ), z ∈ ℝ_{δ_{0}}^{±} .

Let (g_{n})_{n∈ℕ} and (h_{n})_{n∈ℕ} be the sequences of complex valued functions defined on the strip ℝ_{δ_{0}} by

g_{0}(z) := \frac{χ_{+} (z)}{a_{1}(z)} , g_{n+1} (z) := ∑_{j=2}^{q} b_{j} (z)g_{n}(z + β_{j}),
h_{0}(z) := \frac{χ_{−} (z)}{a_{q}(z)} , h_{n+1} (z) := ∑_{j=1}^{q−1} c_{j} (z)h_{n}(z − γ_{j}).

It is clear that all the functions g_{n|ℝ} and h_{n|ℝ} belong to C_{M} {ℝ}.

Let us set

K_{1} := {β_{j} : j = 2, …, q}, K_{2} := {γ_{j} : j = 1, …, q − 1}.

It follows from (2.1) that we have for every n ∈ ℕ, z ∈ ℝ_{δ_{0}}

|g_{n+1} (z)| ⩽ exp(Le^{C| Re(z)|} ) max_{u∈z+K_{1}} |g_{n}(u)|,
|h_{n+1} (z)| ⩽ exp(Le^{C| Re(z)|} ) max_{u∈z−K_{2}} |h_{n}(u)|

where L > 1 is a constant. Then we have for all n ∈ ℕ^{∗}, z ∈ ℝ_{δ_{0}}

|g_{n}(z)| ⩽ exp( ∑^{n−1} ; j=0 Le^{C (| Re(z)|+jβ_{q})} ) max_{u∈z+K_{1}^{(n)}} |g_{0}(u)|

⩽ exp(nLe^{C(| Re(z)|+nβ_{q})} ) max_{u∈z+K_{1}^{(n)}} |χ_{+} (u)|,

|h_{n}(z)| ⩽ exp( ∑_{j=0}^{n−1} Le^{C(| Re(z)|+jγ_{1} |)} ) max_{u∈z−K_{2}^{(n)}} |h_{0}(u)|

⩽ exp(nLe^{C(| Re(z)|+nγ_{1} )} ) max_{u∈z−K_{2}^{(n)}} |χ_{−} (u)|.

Let a > 0. There exists N_{a} ∈ ℕ^{∗} such that (β_{2} + γ_{q−1} )N_{a} ⩾ a and
z + K_{1}^{(n)} ⊂ ℝ_{δ_{0}}^{+} , n ⩾ N_{a}, z ∈ ℝ_{δ_{0} ,a},
z − K_{2}^{(n)} ⊂ ℝ_{δ_{0}}^{−} , n ⩾ N_{a}, z ∈ ℝ_{δ_{0} ,a}.
It follows then from (3.1) that we have for all n ⩾ N_{a}, z ∈ ℝ_{δ_{0} ,a}
max_{u∈z+K_{1}^{(n)}} |χ_{+} (u)| ⩽ D_{1} exp( − cos(C_{1}δ_{0}) exp( C_{1} min_{u∈z+K_{1}^{(n)}} | Re(u)|) )
⩽ D_{1} exp( − cos(C_{1}δ_{0})e^{C_{1} (−a+nβ_{2} )} ),
max_{u∈z−K_{1}^{(n)}} |χ_{−} (u)| ⩽ D_{1} exp( − cos(C_{1}δ_{0}) exp( C_{1} min_{u∈z−K_{1}^{(n)}} | Re(u)|) )
⩽ D_{1} exp( − cos(C_{1}δ_{0})e^{C_{1} (−a+nγ_{q−1} )} ).

Consequently we have for all n ⩾ N_{a}, z ∈ ℝ_{δ_{0} ,a}

|g_{n}(z)| ⩽ D_{1} exp(nLe^{C(a+nβ_{q})} − cos(C_{1}δ_{0})e^{C_{1} (−a+nβ_{2} )} ),
|h_{n}(z)| ⩽ D_{1} exp(nLe^{C(a+nγ_{1} )} − cos(C_{1}δ_{0})e^{C_{1} (−a+nγ_{q−1} )|} ).

On the other hand we have

nLe^{C(a+nβ_{q})} = o_{n→+∞} [cos(C_{1}δ_{0})e^{C_{1} (−a+nβ_{2} )} ],
nLe^{C(a+nγ_{1} )} = o_{n→+∞} [cos(C_{1}δ_{0})e^{C_{1} (−a+nγ_{q−1} )|} ].

So, there exist real constants D_{a} > 0 and E_{a} > 0 and an integer P_{a} ⩾ N_{a} such that the following inequalities hold

|g_{n}(z)| ⩽ D_{a} exp( − E_{a}e^{C_{1} (−a+nβ_{2} )} ), z ∈ ℝ_{δ_{0} ,a}, n ⩾ P_{a},
|h_{n}(z)| ⩽ D_{a} exp( − E_{a}e^{C_{1} (−a+nγ_{q−1} )|} ), z ∈ ℝ_{δ_{0} ,a}, n ⩾ P_{a}.

It follows that the function series ∑ g_{n|ℝ_{δ_{0}}} and ∑ h_{n|ℝ_{δ_{0}}} are uniformly convergent on every compact subset of ℝ_{δ_{0}} and that the functions ∑ _{n=P_{a}}^{+∞} g_{n} and ∑ _{n=P_{a}}^{+∞} h_{n} are holomorphic on ℝ_{δ_{0} ,a} for every a > 0. Let G_{+} and G_{−} be the sums of ∑ g_{n}|_{ℝ_{δ_{0}}} and ∑ h_{n}|_{ℝ_{δ_{0}}} , respectively. Since all the functions g_{n|ℝ} and h_{n|ℝ} belong to C_{M} {ℝ}, it follows that the functions g_{+} := G_{+} |_{ℝ} and g_{−} := G_{−} |_{ℝ} belong to C_{M} {ℝ}. Elementary computations show that

∑_{j=1}^{q} a_{j} (x) g_{+} (x + α_{j}) = χ_{+} (x), x ∈ ℝ,

∑_{j=1}^{q} a_{j} (x) g_{−} (x + α_{j}) = χ_{−} (x), x ∈ ℝ.

Then it follows from (3.1) that the function g := g_{+} + g_{−} is a solution on the interval [−δ_{0}, δ_{0}] of the difference equation (1.1). But the function

x ↦ ∑_{j=1}^{q} a_{j} (x) g(x + α_{j}) − χ(x)

belongs to the quasianalytic Carleman class C_{M} {ℝ}. Consequently the function g ∈ C_{M} {ℝ} is a solution on ℝ of difference equation (1.1). The proof of the main result is complete.

Acknowledgments. I want to express my profound gratitude to professors Ahmed Zeriahi, Lu Hoang Chinh, Jean-Pierre Demailly and Mokhtari Boucif for their great support without which this work would not have been possible. I am also indebted to Professor Alexander Borichev for his precious advices and valuable suggestions. My thanks to the referee for his/her remarks, which have improved this paper.

References

1. G. Belitskii, E. M. Dyn’kin, V. Tkachenko, Difference equations in Carleman classes, Entire functions in modern analysis, Proc. Isr. Math. Conf., 15, 2001, 31-36.

2. G. Belitskii, V. Tkachenko, One dimensional Functional Equations, Operator theory, advances and applications, Vol 114, Birkhauser, 2003.

3. E. M. Dyn’kin, Pseudoanalytic extension of smooth functions. The uniform scale, Trans. Am. Math. Soc., (2) 115, 1980, 33–58.

Department of Mathematics (Received 09 06 2015) Faculty of Sciences (Revised 21 11 2015) Ibn Tofail University Kenitra Morocco

hzoubeir2014@gmail.com