# Does the Fourier series of a continuous function on $\mathbb{T} = [-\pi, \pi]$ converge pointwise or uniformly almost everywhere?

At the end of a chapter on a book on Harmonic Analysis, the author intentionally glosses over many key results in Fourier Series Analysis, concluding:

Question: By "convergence" does the author mean "pointwise convergence"? That is, if a function is continuous, does its Fourier series converge pointwise or uniformly almost everywhere?

• The word convergence appears exactly once in the passage. There it means convergence for all points in a set of full measure, That is, pointwise off of a set of measure zero. – kimchi lover Oct 30 '17 at 14:02
• It is the usual let's get out of here as soon as possible when approaching Carleson's theorem (en.wikipedia.org/wiki/Carleson%27s_theorem), whose proof is very instructive but pretty long and technical. – Jack D'Aurizio Oct 30 '17 at 14:12

The concept of "uniformly almost everywhere" is hardly ever used, and here is why. First, uniform convergence is only practical for continuous functions, such as the partial sums of a Fourier series. Second, suppose there is a set of measure zero $E$ such that continuous functions $\{f_k\}$ converge uniformly on $E^c$. This implies the Cauchy property $$\sup_{E^c}|f_k-f_j| \to 0\quad \text{as } k,j\to\infty$$ But sets of measure zero have empty interior, which together with continuity yield $$\sup |f_k-f_j| = \sup_{E^c}|f_k-f_j|$$ and thus we get uniform convergence everywhere.