ABOUT A CONJECTURE ON DIFFERENCE EQUATIONS IN QUASIANALYTIC CARLEMAN CLASSES

Hicham Zoubeir

Abstract: We consider the difference equation ${\sum }_{j=1}^q{a}_j\left(x\right)\phi (x+{\alpha }_j)=\chi \left(x\right)$ where ${\alpha }_1<\cdots <{\alpha }_q$ ($q\ge 3$) are given real constants, $a_j$ ($j=1,\cdots ,q$) are given holomorphic functions on a strip ${ℝ}_{\delta }$ ($\delta >0$) such that $a_1$ and $a_q$ vanish nowhere on it, and $\chi$ is a function belonging to a quasianalytic Carleman class $C_M\left\{ℝ\right\}$. We prove, under a growth condition on the functions $a_j$, that the difference equation above is solvable in $C_M\left\{ℝ\right\}$.

Keywords: difference equations; quasianalytic Carleman classes

Classification (MSC2000): 30H05; 30B10; 30D05

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