Consider a periodic function . Its Fourier series
always exists formally, but questions of whether its Fourier series converges to itself can be rather subtle. In general,
may be different from its Fourier series: a Fourier coefficient is the evaluation of an integral, so
can be changed pointwise without affecting the value of any Fourier coefficient. It may be natural to believe that convergence holds when
is continuous. However, an example constructed by du Bois-Reymond showed that continuity is not sufficient. The most general result in this area is Carleson’s theorem, showing that
-functions have convergent Fourier series almost everywhere.
Here we will prove that the Fourier series of converges to
in the sense of Cesàro (or Abel), and we will highlight through a Tauberian-type theorem that convergence holds when
is “smooth”. In particular, the main theorem we will show is that for a periodic, continuous function, its Fourier series converges pointwise if its Fourier coefficients are decaying at a linear rate.
![Rendered by QuickLaTeX.com f:[-\pi, \pi] \rightarrow \mathbb{C}](https://www.victorchen.org/wp-content/ql-cache/quicklatex.com-d92594cba58b46ffe36b5f379c56aed7_l3.png)







What types of functions are characterized by decaying Fourier coefficients?
A simple calculation with integration by parts shows that if is differentiable, then
. Since
by the Riemann-Lesbesgue lemma, we have the following:
![Rendered by QuickLaTeX.com f:[-\pi, \pi] \rightarrow \mathbb{C}](https://www.victorchen.org/wp-content/ql-cache/quicklatex.com-d92594cba58b46ffe36b5f379c56aed7_l3.png)

In general, convergence also holds for other types of smoothness conditions.
Summability
We first take a detour (without the Fourier analysis setting) to investigate the different notions of summability for a sequence. The standard and familiar notion of summability captures the intuitive notion that as one adds more numbers from a sequence, the sum gets closer to its limit. More precisely,





Under this definition, the partial sums of Grandi’s series oscillate between
and
, and thus the series diverges. However, one could argue that this series “hovers” around
since the partial sums oscillate between
and
. In fact, the average partial sums of the sequence
converge to
. This motivates the following alternate definition of summability:








It is instructive to rewrite the Cesàro mean as
We can see that the Cesàro mean is a weighted average of the ‘s, where the later terms contribute with a discount linear factor. With this insight, it is intuitive to conclude if a sequence is summable, then it is also Cesàro-summable: a summable sequence must have vanishing “tail”, implying that its Cesàro means form a Cauchy sequence and thus converge.
We can consider another definition of summability where the discount factor is exponential:





Intuitively, Abel summability ought to capture a larger class of sequences. In fact, one can summarize this entire section in the following:













and when is sufficiently large,
,
implying that as
.
Now suppose that converges to
. Define
. Then we can write
Since , using a similar calculation as above, we can write
Since converges (and is thus bounded),
converges for every
and so does
.
Now it remains to show that . Let
, then there exists
such that for each
,
. We can split
into two sums:
Since one can exchange limit with finite sums, is
. So
A similar shows that , and hence
is Abel summable to
.
A Tauberian-type theorem
In the previous section, we showed that the class of summable sequences is a subset of Cesàro-summable sequeences, which in turn is a subset of Abel-summable sequences. It is not difficult to see that these containments are strict. However, by imposing certain conditions, these containments can also be reversed, and statements of this type are called “Tauberian-type theorems”. Here we prove a simple version where the magnitude of the sequence decays linearly.


(a) If




(b) If












Recall that in the previous section, we proved that
Then we have
By assumption, there is some such that for every
,
, implying
. Then when
is sufficiently large,
, proving that
as desired.
(b) Let . Define
. By assumption, when
is sufficiently large,
, and by triangle inequality,
is at most
, so it suffices to prove that
. By triangle inequality again, we have
By assumption, there is some such that for every
,
. Again we can break the the above sum into two, and for the later terms, we have
For the initial terms, we have
where the last inequality is due to Bernoulli’s inequality, and is bounded above by when
is sufficiently large. Together we have
as desired.
Fourier series
We now come back to the Fourier setting. First recall that





We now introduce Dirichlet and Fejér kernels, which when convolved with a function , we obtain the partial sum and Cesàro mean of
, letting us invoke Tauberian Theorem.






![Rendered by QuickLaTeX.com x \in [-\pi, \pi]](https://www.victorchen.org/wp-content/ql-cache/quicklatex.com-5c47b30ff05284dd1662b91e5a529e15_l3.png)










and similarly, the -th Cesàro mean of
is
where the second line follows from the linearity of convolution. Thus, by Tauberian theorem, if converges, then
converges to the same value as well.
To prove the main theorem, it now suffices to show that approaches
. We make one last definition:


(a) for each


(b) for each


(c) for any



The idea is that a family of good kernels essentially approximate the Dirac delta function, which is a single point with infinite mass at and vanishes everywhere else.

We will skip the proof of this fact as it is a straightforward calculation once one establishes the trignometric identity . (However, the Dirichlet kernels cannot be a family of good kernels, otherwise every Fourier series converges to its function, which is not true.) Intuitively, since
is a weighted average of
with highest weights around
(by the definition of a good family of kernels), we can expect that this value is close to
, as long as the
does not change its value too much around
. Now we formalize and prove this intuition, which will finish the proof of our main theorem.





where the first through third lines follow from the definition of convolution, Condition (a) of a good family of kernels, and triangle inequality.
The idea is that when integrating over ,
is small if
is bounded away from
, and
is small if
is close to
. Formally, let
. Since
is continuous at
, there is some
such that
for some by Condition (b) of a good family of kernels. Since
can be arbitrary, the above integral is
.
Now for the other values of , note that since
is integrable, the function is bounded by some
. Then
which tends to as
, finishing the proof.


