# Exponential sum and the equidistribution theorem

Consider the sequence , where gives the fractional part of a real number. The equidistribution theorem states that is irrational iff for any sub-interval of the unit interval, for sufficiently large , roughly of the numbers fall inside . More precisely,

(Equidistributed sequence) A sequence with is equidistributed mod 1 if for every ,

(Equidistribution theorem) The sequence is equidistributed mod 1 iff is irrational.

Let us try to understand why the sequence behaves differently depending on whether is rational or not. If for some integers and , then roughly of the first multiples of are integers, i.e., . In fact, it is not difficult to see that for , fraction of the sequence equals . So equidistribution does not occur when is rational.

Now for , consider the exponential function , which has period . As discussed above, roughly of the sequence is , so the exponential sum contains an infinite number of ones. For example if , the exponential sum evaluates to , which is Grandi’s series and diverges. For an arbitrary , the exponential sum is a repeating sum of all -th roots of unity, and one can use the same technique showing that Grandi’s series diverges to establish that this exponential sum diverges too. However, the case when is irrational differs: every nonzero integer multiple of is an non-integer, and thus for any . Writing , the exponential sum converges to the geometric series , since .

In fact, the above intuition can be summarized into the following criterion due to Weyl:

(Weyl’s criterion) A sequence is equidistributed mod 1 iff for all integers ,

From the above discussion, Weyl’s criterion immediately implies the equidistribution theorem.

(of equidistribution theorem) Suppose is rational, i.e., for some integers and . Without loss of generality, we assume . Then for all integer , by Euclid’s algorithm, is in . So the sequence is not equidistributed, e.g., no multiples of fall inside .

Now suppose is irrational. Since , we have

which tends to as .

Before proving Weyl’s criterion, it is instructive to state the notion of equidistribution analytically. For , let iff . Note that is a periodic function on all of .

(Weyl’s criterion restated) Let be a sequence in . The following two statements are equivalent:
1) For ,

2) For all integers ,

Written in this form, Weyl’s criterion is asserting that if the sum of every interval function at a sequence converges to its integral, then every exponential sum converges to its integral as well, and vice versa. The reason is that both the space of interval functions and the space of exponential functions can approximate one another. More precisely, an exponential function is clearly periodic and continuous (and thus uniformly continuous), and every uniformly continuous function can be approximated by a finite sum of interval functions. For the other direction, a periodic, interval function can be approximated by a periodic, continuous function, which by Fejér’s Theorem can also be approximated by a finite sum of exponential functions. Now we have all the machinery to write down a proof.

(of Weyl’s criterion) For the forward direction (1) => (2), fix and . Since the exponential function continuous on the closed interval , it is also uniformly continuous, i.e., there exists some such that for every , . Let be a step function where if . By construction, is a finite sum of interval functions with .

By linearity, we have

By taking sufficiently large and repeatedly applying triangle inequality, we have

implying that converges to as .

Now we consider the reverse direction (2) => (1). Fix and let . Define

Note that is continuous, periodic, and dominates . By Fejér’s Theorem, there exists such that . By linearity, we have

By taking sufficiently large and repeatedly applying triangle inequality, we have

implying that

Since dominates , we have

One can similarly define a periodic, continuous function that approximates from below, showing that is bounded below by , which together with the proceeding argument will show that the limit of must exist and converge to .

# Summability, Tauberian theorems, and Fourier series

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.

If and and is continuous at , then as .
The sufficient condition in the above theorem can be improved to , but we won’t prove the stronger statement here.

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:

If is differentiable, then its Fourier series converges pointwise to .

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,

A sequence is summable to if its sequence of partial sums where converges to .

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:

A sequence is Cesàro-summable to if its sequence of Cesàro means converges to , where and is the -th partial sum .

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:

A sequence is Abel summable to if for every , converges and .

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

If a sequence is summable to , then it is Cesàro-summable to . If a sequence is Cesàro-summable to , then it is also Abel-summable to .
Suppose first that is summable to . For every , there exists such that for every , . Let and . Then for every , we have

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.

(Tauberian) Suppose the sequence satisfies .
(a) If is Cesàro-summable to , then is also summable to .
(b) If is Abel-summable to , then is also summable to .
(a) Fix . Let be the Cesàro mean and partial sum of , respectively. By assumption, when is sufficiently large, and by triangle inequality, is at most , so it suffices to prove that .

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

For a periodic function , we define its -th partial Fourier sum to be , with , and .

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.

is the -th Dirichlet kernel, and is the -th Fejér kernel.
(Tauberian theorem restated) Let be a periodic function with . Then for any , if for some as , then as .
For any nonnegative integer , let . Note that by assumption . By the Convolution Theorem, the -th partial sum of is simply

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:

Let be a periodic function. We say that is a family of good kernels if the following three conditions all hold:
(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.

The Fejér kernels is a family of good kernels.

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.

Suppose is a family of good kernels. Let be a periodic, integrable function that is continuous at . Then as .
We first write

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.

In the previous section, we showed that the Abel mean can play a similar role to Cesàro. Similarly, one can define the Poisson kernel and show that it is a good family of kernels as to prove the main theorem through instead.

# The Basel problem via Fourier series

The Basel problem asks to compute the summation of the reciprocal of the natural numbers squared, and it is known that

While there are numerous known proofs (our presentation resembles Proof 9 most closely), we give a self-contained one that motivates the usage of Fourier analysis for periodic functions.

Recall that a function is -periodic if for every , . From Fourier analysis, every such function can be expressed as a series of cosines and sines. More precisely,

If is continuous and -periodic, then

where , and is the -th Fourier coefficient equal to .

We say that is the Fourier series of . When we define the inner product of two functions to be , then it is easily seen that the “characters” form an orthonormal system, and the -th Fourier coefficient can be viewed as the projection of along the -th character. The Fourier theorem asserts that can be approximated in -norm by its Fourier series. In fact, by applying the Weierstrass M-test, we can make the following statement with a stronger convergence guarantee:

If is continuous and -periodic and is convergent, then its Fourier series converges uniformly to .

Furthermore, the Fourier transform itself encodes many of the properties of the original function . In particular, we note that the symmetry of is preserved in its Fourier transform.

If is even, then for every , .
If is odd, then for every , .

Now we have the machinery to compute .

(Of Basel) Define to be . We compute the Fourier coefficient of . First note that

For , we have

Using integration by parts, we have for every ,

Then

Thus,

From Euler’s formula, the Fourier series of can be written as

Since is even, by Proposition, . From our calculation, the Fourier series of is explicitly

Note that is convergent from the Cauchy condensation test. Thus, by the Convergence Theorem, we conclude that

where the convergence is uniform (and in particular, pointwise). Setting , we see that

Lastly, observe that

finishing the proof.