Let $Q$ be a bounded domain in $\mathbf R^N$ 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 
$$
\begin{cases}
-\Delta \psi_k= \lambda_k \psi_k,&\text{ in } Q,\\  
\dfrac{\partial \psi_k}{\partial n}=0,&\text{ on }\partial \Omega.
\end{cases}$$ 
* Finally denote 
$$
S_n(x)=\sum\limits_{k=0}^{n}\int_Q f\psi_k\operatorname{d\!}x\, \psi_k(x).
$$ 

Under what (additional) conditions we can expect 
$$
\lim\limits_{n\to \infty}\Vert f-S_n\Vert_{C(\overline{Q})}=0\; ?
$$