Integration of Fourier Series

When we deal with Fourier series, a crucial aspect is their convergence. While a Fourier series might not always converge at every point, integrating the Fourier series term by term is a powerful technique that allows us to recover the original function or a related function, often with improved convergence properties. This process relies on the fact that integration is a continuous operation.

Consider a function $f(x)$ that is periodic with period $2\pi$. Its Fourier series is given by:

$$f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))$$ where the coefficients are: $$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \cos(nt) dt, \quad n \ge 0$$ $$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \sin(nt) dt, \quad n \ge 1$$

If we integrate this Fourier series term by term from $-\pi$ to $x$, we get:

$$\int_{-\pi}^{x} f(t) dt \sim \int_{-\pi}^{x} \frac{a_0}{2} dt + \sum_{n=1}^{\infty} \int_{-\pi}^{x} (a_n \cos(nt) + b_n \sin(nt)) dt$$

Let's evaluate the integrals:

The integral of the constant term is:

$$\int_{-\pi}^{x} \frac{a_0}{2} dt = \frac{a_0}{2} [t]_{-\pi}^{x} = \frac{a_0}{2} (x - (-\pi)) = \frac{a_0}{2} (x + \pi)$$

The integral of the cosine term is:

$$\int_{-\pi}^{x} a_n \cos(nt) dt = a_n \left[\frac{\sin(nt)}{n}\right]_{-\pi}^{x} = a_n \left(\frac{\sin(nx)}{n} - \frac{\sin(-n\pi)}{n}\right) = \frac{a_n}{n} \sin(nx)$$

The integral of the sine term is:

$$\int_{-\pi}^{x} b_n \sin(nt) dt = b_n \left[-\frac{\cos(nt)}{n}\right]_{-\pi}^{x} = b_n \left(-\frac{\cos(nx)}{n} - (-\frac{\cos(-n\pi)}{n})\right) = -\frac{b_n}{n} (\cos(nx) - \cos(n\pi))$$

So, the integrated series becomes:

$$\int_{-\pi}^{x} f(t) dt \sim \frac{a_0}{2} (x + \pi) + \sum_{n=1}^{\infty} \frac{a_n}{n} \sin(nx) + \sum_{n=1}^{\infty} -\frac{b_n}{n} (\cos(nx) - \cos(n\pi))$$

This process of term-by-term integration is valid if the original function $f(x)$ is integrable over $[-\pi, \pi]$, which is generally true for functions with a Fourier series. The resulting series represents the integral of $f(x)$.

An important consequence of integrating Fourier series is that it can establish the convergence of the series under weaker conditions. For instance, if $f(x)$ is continuous and has a piece-wise continuous derivative, its Fourier series converges everywhere. However, integration can help us deal with functions that might not meet these strict criteria.

The integration of Fourier series is closely related to the concept of Cesàro summability and the behavior of Fourier series at points of discontinuity or for functions that are not necessarily continuous everywhere.

Fejér's Theorem on Cesàro Summability at a Point

Fejér's theorem is a fundamental result in Fourier analysis that provides a strong condition for the convergence of the Cesàro means of a Fourier series. The Cesàro mean provides a way to average the partial sums of a series, which can help in cases where the series itself does not converge.

Let $S_N(x) = \frac{a_0}{2} + \sum_{n=1}^{N} (a_n \cos(nx) + b_n \sin(nx))$ be the $N$-th partial sum of the Fourier series of $f(x)$. The Cesàro mean of order 1, denoted by $\sigma_N(x)$, is the average of the first $N+1$ partial sums ($S_0, S_1, \ldots, S_N$):

$$\sigma_N(x) = \frac{1}{N+1} \sum_{k=0}^{N} S_k(x)$$

Fejér's theorem states that if $f(x)$ is a periodic function (with period $2\pi$) that is integrable on $[-\pi, \pi]$, and if $x_0$ is a point where $f(x)$ is continuous, then the Cesàro means $\sigma_N(x_0)$ converge to $f(x_0)$ as $N \to \infty$.

More formally:

This theorem is significant because it establishes convergence for a broader class of functions than simple pointwise convergence of the Fourier series itself. For instance, a continuous function might have a Fourier series that does not converge pointwise at certain points, but its Cesàro means will always converge to the function's value at those points.

The proof of Fejér's theorem involves expressing $\sigma_N(x)$ in terms of $f(x)$ and a kernel function, known as the Fejér kernel, $F_N(x)$:

$$\sigma_N(x) = \frac{1}{2\pi(N+1)} \int_{-\pi}^{\pi} f(x+t) \left( \frac{\sin((N+1)t/2)}{\sin(t/2)} \right)^2 dt$$

The Fejér kernel, $F_N(t) = \frac{1}{2\pi(N+1)} \left( \frac{\sin((N+1)t/2)}{\sin(t/2)} \right)^2$, has important properties:

• $F_N(t) \ge 0$ for all $t$.
• $\frac{1}{2\pi} \int_{-\pi}^{\pi} F_N(t) dt = 1$.
• For any $\delta > 0$, $\max_{|t| \ge \delta} |F_N(t)| \to 0$ as $N \to \infty$.

Using these properties, one can show that if $f$ is continuous at $x_0$, then $\sigma_N(x_0) \to f(x_0)$. The non-negativity of the Fejér kernel is crucial for this convergence.

Example: Consider the function $f(x) = x$ for $-\pi < x < \pi$, and $f(x+2\pi) = f(x)$. This function is continuous everywhere except at $x = \pi, -\pi, 3\pi, \ldots$. Its Fourier series is $f(x) \sim 2 \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(nx)$. At $x = \pi/2$, $f(\pi/2) = \pi/2$. Fejér's theorem guarantees that the Cesàro means of this series at $x = \pi/2$ will converge to $\pi/2$.

Fejér–Lebesgue Theorem on Summability Almost Everywhere

While Fejér's theorem guarantees convergence of Cesàro means at points of continuity, the Fejér–Lebesgue theorem extends this concept to almost every point, even for functions that might not be continuous everywhere. This is a more general result that highlights the power of Cesàro summability.

The theorem states that for any $2\pi$-periodic function $f(x)$ that is Lebesgue integrable on $[-\pi, \pi]$, its Fourier series is Cesàro summable to $f(x)$ almost everywhere (a.e.). This means that the Cesàro means $\sigma_N(x)$ converge to $f(x)$ for all $x$ except possibly on a set of Lebesgue measure zero.

Formally:

This is a profound result because it implies that even for functions with many discontinuities (like the Dirichlet function, which is not Riemann integrable but is Lebesgue integrable), the Cesàro means of their Fourier series will still "find" the function's value almost everywhere.

Key implications:

• Universality of Cesàro Summability: It shows that Cesàro summability is a very robust form of convergence for Fourier series.
• Integrability and Convergence: It connects the property of Lebesgue integrability of a function directly to the Cesàro summability of its Fourier series almost everywhere.
• Improvement over Pointwise Convergence: It is a stronger statement than pointwise convergence of the Fourier series itself, which might fail even for continuous functions.

The proof of the Fejér–Lebesgue theorem relies on the properties of the Fejér kernel and the Lebesgue integral. It essentially uses the same machinery as the proof of Fejér's theorem but applies it in the context of the more general Lebesgue integral, allowing for sets of measure zero where the convergence might not hold.

Example: The Dirichlet Function

Consider the Dirichlet function:

$$D(x) = \begin{cases} 1 & \text{if } x \text{ is rational} \\ 0 & \text{if } x \text{ is irrational} \end{cases}$$

This function is periodic with period $2\pi$. It is not Riemann integrable because the upper and lower Riemann sums do not converge to the same value. However, it is Lebesgue integrable. The Lebesgue integral of $D(x)$ over $[-\pi, \pi]$ is 0, since the set of rational numbers has Lebesgue measure zero.

The Fourier series of the Dirichlet function can be shown to be identically zero. The Fejér–Lebesgue theorem states that the Cesàro means of this zero Fourier series will converge to $D(x)$ almost everywhere. Since $D(x)$ is 1 on a set of positive measure (the rationals) and 0 on a set of positive measure (the irrationals), and its integral is 0, this seems paradoxical. The resolution is that the theorem states convergence to $f(x)$ *almost everywhere*. For the Dirichlet function, its Fourier series is identically zero, and its Cesàro means are also identically zero. So, $\sigma_N(x) = 0$ for all $N$. This converges to 0 for all $x$. The function $f(x)$ itself is 1 for rationals and 0 for irrationals. Thus, $\sigma_N(x)$ converges to $f(x)$ only on the set of irrational numbers, which is almost everywhere.

The Fejér–Lebesgue theorem implies that for any Lebesgue integrable function $f$, its Fourier series is Cesàro summable almost everywhere. This is a powerful statement about the behavior of Fourier series.

Relationship between Integration, Fejér's Theorem, and Fejér–Lebesgue Theorem

These concepts are deeply interconnected in Fourier analysis. The ability to integrate Fourier series term by term provides a way to construct new series with potentially better convergence properties. Fejér's theorem then shows that for continuous functions, the Cesàro means of the Fourier series provide a reliable way to recover the function's value at that point. The Fejér–Lebesgue theorem generalizes this by showing that Cesàro summability holds almost everywhere for all Lebesgue integrable functions.

The integration process can be seen as a step towards smoothing out discontinuities. Cesàro summability, as described by Fejér's theorems, provides a form of "averaged" convergence that is more resilient to irregularities in the function than the standard pointwise convergence of the Fourier series itself.

The Fejér kernel's properties are central to proving both Fejér's theorem and the Fejér–Lebesgue theorem. Its non-negativity and its tendency to concentrate around $t=0$ as $N \to \infty$ are key.

In essence, these theorems collectively demonstrate that while a Fourier series might not always converge in the traditional sense, its associated Cesàro means offer a strong form of convergence that holds universally for all integrable functions, at least almost everywhere. This makes Fourier series a remarkably powerful tool for representing and analyzing functions, even those with complex behavior.