# Uniform convergence of parametrised Fourier series.

Let $$X$$ be a compact topological space, and $$f: S^1 \times X \rightarrow \mathbb{C}$$ be a continuous function. Then for every point $$x \in X$$ we can compute the Fourier coefficients $$c_k(x) = \int_{S^1} f(z,x) z^k \text{d}z,$$ and construct the Fourier series as the limit of the partial sums $$s_n(z, x) = \sum_{k=-n}^n c_k z^k.$$ This must not converge point-wise, but taking Cesàro means $$\sigma_l(z,x) = \frac 1 l\sum_{n = 0}^{l-1} s_n(z,x) = \frac 1 l \sum_{n=0}^{l-1}\sum_{k=-n}^{n}c_kz^k = \sum_{k=-l}^l\frac{l - |k|}{l}c_kz^k,$$ we get uniform convergence in $$z$$ by Fejér's theorem.

Note that $$\sigma_l(z,x)$$ is still continuous in $$x$$.

In his book K-Theory, Atiyah claims that the $$\sigma_l$$ converge uniformly in $$x$$, that is given $$\epsilon > 0$$ there exists $$l_0 \in \mathbb{N}$$, such that for all $$(z,x) \in S^1 \times X$$ and $$l \geq l_0$$ we have $$\|f(z,x) - \sigma_l(z,x)\| < \epsilon .$$

I don't have any idea why this should be true, but my knowledge about Fourier series and analysis is pretty limited. Any help would be appreciated.

Here are some thoughts I had:

• One obstacle is that the $$\sigma_l$$ do not form a true series, because of the factor $$\frac 1 l$$. So the classical Weierstrass test does not even apply here. Is there another kind of Weierstrass test that one could apply to this situation?

• Because $$X$$ is compact, it would suffice to show uniform convergence locally.