# Approximation of Hölder functions by Fourier series

Let $$Q$$ be a bounded domain in $$\mathbf R^N$$ with smooth boundary. Let $$f\in C^a(\overline{Q})$$, $$0.

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

• I cerrected. Thank you. Feb 26 at 5:22
• This is not true in higher dimension, basicly because the sequence of eigenfunctions need not be uniformly bounded. An example of this situation is the Laplace Beltrami on the unit sphere $S^{N-1}$ of $\mathbb R^N$. The eigenfunctions are the spherical harmonics, if $k$ is the degree, the supnorm is like $k^{N-2}$ and the expansion converges in the supnorm if $f \in C^l$ with $l>(N-1)/2$. Feb 26 at 8:36
• Please, quote my name if you reply to me, to let me receive a notification. If $-\Delta \psi_k=\lambda_k \psi_k$ then $(-\Delta)^\ell \psi_k=\lambda_k^\ell \psi_k$ and by elliptic estimates the norm of $\psi_k$ in the Sobolev space $H^{2\ell}$ is boounded by $\lambda_k^\ell$. Since $\lambda_k \approx k^{2/N}$ choosing $2\ell >N/2$ by Sobolev embedding you get a rough estimate in the sup norm. I am counting eigenvalues and eigenfunctions with multiplicity (in the comment before all spherical harmonics of degrre $k$ belong to the same $\lambda_k$). Feb 26 at 15:31
• @WillieWong One argument is short but uses elliptic regularity. For $\psi \in C^{2\ell}$ write $\psi=\sum_k (\psi, \psi_k)\psi_k$ in $L^2$ and $$\Delta^\ell \psi=\sum_k (\Delta^\ell \psi, \psi_k)\psi_k=\sum_k (\psi, \Delta^\ell \psi_k)\psi_k=\sum_k \lambda^\ell_k ( \psi, \psi_k)\psi_k=\Delta^\ell (\sum_k (\psi, \psi_k)\psi_k).$$ This gives that the series yielding $\psi$ converges in $H^{2\ell}$, by elliptic regularity, and by Sobolev embedding uniformly, for $\ell$ large. Feb 27 at 18:25
• @WillieWong The other one I know uses more on spherical harmonics, zonals, sup-norm estimates. If you need, I can write down it. Feb 27 at 18:27

Hölder functions periodic with period 1 satisfy the Dini criterion $$\int_0^{1/2}\frac{\vert u(x+t)- u(x)\vert}{t} dt<+\infty,$$ thus their Fourier series are uniformly convergent (towards $$u$$). On the other hand the Besov space $$B^\alpha_{\infty, \infty}$$ is equal to the Hölder space $$C^\alpha$$ for $$\alpha\in (0,1)$$.