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

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...

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 $$\mathbb{S}^2$$.
$$f(t)=\sum\limits_{m}^\infty \frac{1}{n^2}\varphi_{\lambda_n}(\lambda_nt),$$
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).
• It might be interesting to plot (approximately) such a function and the partial sums of its Fourier Series. The $\lambda_n$ are so large however that it does not look very feasible. Oct 22, 2019 at 20:58