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.
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:
In general, convergence also holds for other types of smoothness conditions.
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 is Cesàro-summable to , then is also summable to .
(b) If is Abel-summable to , then is also summable to .
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.
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.
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 , , and
(c) for any , as .
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.