Let \(\omega(r)\) be an increasing function defined for \(0\leq r<1\) such that \(\lim_{r\to 1}\omega(r)=\infty\). In a paper of J. P. Kahane and Y. Katznelson and in a paper of M. Ortel and W. Schneider, there is proved that for every measurable function \(\phi\) on the boundary of the unit disc D, there exists a holomorphic function \(f\) in \(D\) such that \(| f(z)| \leq \omega (| z|)\) for every \(z\in D\) and \(\lim_{r\to 1}f(r\zeta)=\phi (\zeta)\) almost everywhere on \(\partial D.\)
In this paper we prove that for every measurable function \(\phi\) on the boundary of the unit ball \(B\) of \({\mathbb{C}}^ n\) there exist an increasing sequence \(\{r_ j\}\), \(\lim_{j\to \infty} r_ j=1\) and a holomorphic function \(f\) in \(B\) such that \(| f(z)| \leq \omega (| z|)\) for every \(z\in B\) and \(\lim_{n\to \infty} f_{r_ n}(\zeta)=\phi (\zeta)\) almost everywhere on \(\partial B\). As a consequence we obtain that for every measurable function \(\phi\) on \(\partial B\) there exists a holomorphic function \(f\) in \(B\), \(f\in \cap_{1\leq p<\infty}L^ p(B)\) such that \(\phi\) (\(\zeta)\) belongs to the cluster set of \(f\) almost everywhere on \(\partial B\).