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:

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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?

  • $\begingroup$ 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. $\endgroup$ – kimchi lover Oct 30 '17 at 14:02
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    $\begingroup$ 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. $\endgroup$ – Jack D'Aurizio Oct 30 '17 at 14:12

It means "pointwise, almost everywhere".

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.

Aside: there is a useful notion of convergence "uniformly, except on a set of arbitrarily small measure", which appears in Egorov's theorem. But that is not the same as "almost everywhere".


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