The Cauchy-Riemann equations for a class of $(0,1)$-forms in $l^{2}$

We study the local exactness of $\overline{\partial}$ operator in the unit ball of $l^2$ for a particular class of $(0,1)$-forms $\omega$ of the type $\omega (z) = \sum_i z_i\omega^i(z) d\overline{z_i}$, $z = (z_i)$ in $l^2$. We suppose each function $\omega^i(z)$ of class $C^{\infty}$ in the closed unit ball of $l^2$ of the form $\omega^i(z) = \sum_k \omega^i_k\big(z^k\big)$, where $\mathbb{N} = \bigcup I_k$ is a partition of $\mathbb{N},$ ($\textrm{card} I_k)\leq+\infty$, and $z^k$ is the projection of $z$ on $\mathbb{C}^{I_k}$. We establish sufficient conditions for exactness of $\omega$ related to the expansion in Fourier series of the functions $\omega^i_k$.