We study the local exactness of the ∂¯\documentclass[12pt]{minimal}
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\begin{document}$$\overline{\partial }$$\end{document} operator in the Hilbert space l2\documentclass[12pt]{minimal}
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\begin{document}$$l^2$$\end{document} for a particular class of (0,1)\documentclass[12pt]{minimal}
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\begin{document}$$(0,1)$$\end{document}-forms ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} of the type ω(z)=∑iziωi(z)dz¯i\documentclass[12pt]{minimal}
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\begin{document}$$\omega (z) = \sum _i z_i\omega ^i(z) d\overline{z}_i$$\end{document}, z=(zi)\documentclass[12pt]{minimal}
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\begin{document}$$z = (z_i)$$\end{document} in l2\documentclass[12pt]{minimal}
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\begin{document}$$l^2$$\end{document}. We suppose each function ωi\documentclass[12pt]{minimal}
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\begin{document}$$\omega ^i$$\end{document} of class C∞\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty $$\end{document} in the closed unit ball of l2\documentclass[12pt]{minimal}
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\begin{document}$$l^2$$\end{document}, of the form ωi(z)=∑kωkizk\documentclass[12pt]{minimal}
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\begin{document}$$\omega ^i(z) = \sum _k \omega ^i_k\left( z^k\right) $$\end{document}, where N=⋃Ik\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf N = \bigcup I_k$$\end{document} is a partition of N\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf N$$\end{document}, (\documentclass[12pt]{minimal}
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\begin{document}$$($$\end{document}card Ik<+∞)\documentclass[12pt]{minimal}
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\begin{document}$$I_k < +\infty )$$\end{document} and zk\documentclass[12pt]{minimal}
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\begin{document}$$z^k$$\end{document} is the projection of z\documentclass[12pt]{minimal}
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\begin{document}$$z$$\end{document} on CIk\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf C^{I_k}$$\end{document}. We establish sufficient conditions for exactness of ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} related to the expansion in Fourier series of the functions ωki\documentclass[12pt]{minimal}
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\begin{document}$$\omega ^i_k$$\end{document}.
