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 ($k=0,1,2\dotsc$, $\lambda_0=0$, $\psi_0=1/|Q|$) 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 $$ \lim\limits_{n\to \infty}\lVert f-S_n\rVert_{C(\overline{Q})}=0\; ? $$

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 ($k=0,1,2\dotsc$, $\lambda_0=0$, $\psi_0=1/\sqrt{|Q|}$) 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 $$ \lim\limits_{n\to \infty}\lVert f-S_n\rVert_{C(\overline{Q})}=0\; ? $$

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 $$ \lim\limits_{n\to \infty}\Vert f-S_n\Vert_{C(\overline{Q})}=0\; ? $$

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 ($k=0,1,2\dotsc$, $\lambda_0=0$, $\psi_0=1/|Q|$) 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 $$ \lim\limits_{n\to \infty}\lVert f-S_n\rVert_{C(\overline{Q})}=0\; ? $$