Everywhere Divergent $\Phi$-Means of Fourier Series

For a function $f\in L^1({\mathbb T})$, we investigate the sequence $(C,1)$ of mean values $\Phi(|S_k(x,f)-f(x)|)$, where $\Phi (t)\colon [0,+\infty)\to [0,+\nobreak \infty)$, $\Phi (0)=\nobreak 0$, is a continuous increasing function. We prove that if $\Phi $ increases faster than exponentially, then these means can diverge everywhere. Divergence almost everywhere of such means was established earlier.