Let $Q$ be a bounded domain in $R^N$ with smooth boundary. Let $f\in C^a(\overline{Q})$, $0<a<1$. Denote $\psi_k(x)$ normalized eigenfunctions and $\lambda_k$ eigenvalues ($k=0,1,2...$, $\lambda_0=0$, $\psi_0=1/|Q|$) such that $$-\Delta \psi_k= \lambda_k \psi_k, in \, Q,$$ $$\frac{\partial \psi_k}{\partial n}=0, on \, \partial \Omega.$$ Denote $S_n(x)=\sum\limits_{k=0}^{n}\int_Q f\psi_kdx\, \psi_k(x)$. Under what (additional) conditions we can expect $$ \lim\limits_{n\to \infty}\Vert f-S_n\Vert_{C(\overline{Q})}=0 ? $$