Counterexample to uniform convergence of Laplace series (expansion in spherical harmonics)

Question

Expansion of a real function on 2-sphere in spherical harmonics, so-called Laplace series, converges uniformly for continuously differentiable functions (see e.g. https://projecteuclid.org/euclid.bbms/1103408694).

Is there any explicit example of a real function on 2-sphere, which is merely continuous, and for which the corresponding Laplace series does not converge uniformly? I would expect that this is some "fractal beast", but I might be wrong...

Answer 1

One can just adapt an example from the circle to the sphere by lifting such a function from $\mathbb{S}^1$ to zonal spherical harmonics on $\mathbb{S}^2$.

It is then sufficient to give an example of a continuous function with Fourier series which does not converge pointwise, much less uniformly. On the circle, take the example from Katznelson's Introduction to Harmonic Analysis (part B in Theorem 2.1, Ch. 2),

$$f(t)=\sum\limits_{m}^\infty \frac{1}{n^2}\varphi_{\lambda_n}(\lambda_nt),$$

where $\varphi_n=\sigma_{n^2}(\psi_n,t)$ is a trigonometric polynomial of degree $n^2$, $\sigma_n(f,t)$ is the Fejér sum of $f$ to order $n$, and $\psi_n$ are a sequence of continuous functions satisfying $$\|\psi_n\|_{\infty}\leq 1,\ |S_n(\psi_n,0)|>\frac{1}{2}\|D_n\|_{L^1}>\frac{1}{10}\log{n},$$ where $S_n(f,t)$ is the $n$th partial sum of the F.S. of $f$ evaluated at $t$ and $D_n$ is the Dirichlet kernel. Taking $\lambda_{n}=2^{3^n}$ gives a continuous function with Fourier series diverging at $t=0$. Continuity is checked since the convergence of the series defining $f$ is uniform. Divergence of $f$ at $0$ comes since the partial sums of order $\lambda_n^2$ at $t=0$ grow faster than $\frac{K}{n^2}\log{\lambda_n}-3$ which goes to infinity as $n\rightarrow\infty$.

The functions $\psi_n$ above can be constructed by letting them satisfy $\psi_n(t)=\text{sgn}(D_n(t))$ except in small intervals around the discontinuities of $\text{sgn}(D_n(t))$ (as the preceding section in Katznelson details).