Riesz–Fischer Theorem

The Riesz–Fischer theorem is a fundamental result in the theory of Fourier series. It establishes the existence of Fourier series for a broad class of functions, specifically those that are square-integrable. In simpler terms, it assures us that if a function is "well-behaved enough" (in this case, having a finite integral of its square), then its Fourier series will converge to the function itself in a specific sense. This theorem is crucial because it guarantees that the process of finding Fourier coefficients and constructing a Fourier series is meaningful for a wide range of practical applications.

Statement of the Riesz–Fischer Theorem

Let $f(x)$ be a function defined on the interval $[-\pi, \pi]$ such that $\int_{-\pi}^{\pi} |f(x)|^2 dx < \infty$. This condition means that $f(x)$ is a square-integrable function on the interval. The theorem states that the Fourier series of $f(x)$, given by:

$a_0/2 + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))$

where the Fourier coefficients $a_n$ and $b_n$ are defined as:

$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx$, for $n = 0, 1, 2, \dots$

$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) dx$, for $n = 1, 2, 3, \dots$

converges to $f(x)$ in the mean square sense. This means that the limit of the integral of the square of the difference between $f(x)$ and its partial Fourier sums goes to zero as the number of terms increases. Mathematically, if $S_N(x)$ is the $N$-th partial sum of the Fourier series, then:

$\lim_{N \to \infty} \int_{-\pi}^{\pi} |f(x) - S_N(x)|^2 dx = 0$

Significance of the Riesz–Fischer Theorem

Before the Riesz–Fischer theorem, the convergence properties of Fourier series were not fully understood for all types of functions. This theorem provides a rigorous foundation for using Fourier series to represent functions that are not necessarily continuous or even piecewise continuous. It connects the concept of square-integrability, which is a measure of the "size" of a function in a Hilbert space (specifically, the space $L^2[-\pi, \pi]$), with the existence and convergence of its Fourier series representation. This is immensely important in areas like signal processing, where signals are often represented by functions that might have discontinuities.

Proof Idea (Conceptual)

The proof of the Riesz–Fischer theorem typically involves constructing the Fourier series coefficients and then showing that the sequence of partial sums forms a Cauchy sequence in the $L^2$ space. Since $L^2$ is a complete space, a Cauchy sequence must converge to an element in the space, which is then shown to be the Fourier series of the original function.

Bessel's Inequality

Bessel's inequality is another cornerstone result related to Fourier series and orthogonal function systems. It provides an upper bound on the sum of the squares of the Fourier coefficients of a function. Essentially, it tells us that the "energy" of the function, when decomposed into its Fourier components, is distributed among these components, and the total energy is finite. This inequality is a precursor to Parseval's theorem, which is a stronger statement about the equality of energy.

Statement of Bessel's Inequality

Let $f(x)$ be a square-integrable function on $[-\pi, \pi]$. Its Fourier coefficients are given by:

$a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx$

$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx$, for $n \ge 1$

$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) dx$, for $n \ge 1$

Bessel's inequality states that:

$\frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2) \le \frac{1}{\pi} \int_{-\pi}^{\pi} |f(x)|^2 dx$

This inequality holds for any square-integrable function $f(x)$. The left side represents the sum of the squares of the Fourier coefficients (scaled appropriately), and the right side represents a measure of the total "energy" of the function $f(x)$.

Interpretation of Bessel's Inequality

The inequality tells us that the sum of the squares of the Fourier coefficients is finite. This implies that the Fourier coefficients must approach zero as $n$ approaches infinity. This is known as the Riemann–Lebesgue lemma, which is a direct consequence of Bessel's inequality. The lemma states that for any square-integrable function $f$, $\lim_{n \to \infty} \int_{-\pi}^{\pi} f(x) \cos(nx) dx = 0$ and $\lim_{n \to \infty} \int_{-\pi}^{\pi} f(x) \sin(nx) dx = 0$. In simpler terms, higher frequency components of a function have smaller amplitudes.

Proof Idea (Conceptual)

The proof of Bessel's inequality involves considering the integral of the square of the difference between the function $f(x)$ and its $N$-th partial Fourier sum, $S_N(x)$.

Let $f(x)$ be the function and $S_N(x) = \frac{a_0}{2} + \sum_{n=1}^{N} (a_n \cos(nx) + b_n \sin(nx))$ be its $N$-th partial Fourier sum.

We examine the integral $\int_{-\pi}^{\pi} |f(x) - S_N(x)|^2 dx$.

Expanding this integral and using the orthogonality properties of the trigonometric functions (e.g., $\int_{-\pi}^{\pi} \cos^2(nx) dx = \pi$, $\int_{-\pi}^{\pi} \sin^2(nx) dx = \pi$, $\int_{-\pi}^{\pi} \cos(mx)\cos(nx) dx = 0$ for $m \ne n$, etc., and $\int_{-\pi}^{\pi} \cos(nx) \sin(mx) dx = 0$), along with the definitions of $a_n$ and $b_n$, we can show that:

$\int_{-\pi}^{\pi} |f(x) - S_N(x)|^2 dx = \int_{-\pi}^{\pi} |f(x)|^2 dx - \pi \left( \frac{a_0^2}{2} + \sum_{n=1}^{N} (a_n^2 + b_n^2) \right)$

Since the left side of this equation, $\int_{-\pi}^{\pi} |f(x) - S_N(x)|^2 dx$, must be non-negative (as it's an integral of a squared real quantity), we have:

$\int_{-\pi}^{\pi} |f(x)|^2 dx - \pi \left( \frac{a_0^2}{2} + \sum_{n=1}^{N} (a_n^2 + b_n^2) \right) \ge 0$

Rearranging this inequality gives:

$\frac{a_0^2}{2} + \sum_{n=1}^{N} (a_n^2 + b_n^2) \le \frac{1}{\pi} \int_{-\pi}^{\pi} |f(x)|^2 dx$

Since this inequality holds for any $N$, we can take the limit as $N \to \infty$ to obtain Bessel's inequality:

$\frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2) \le \frac{1}{\pi} \int_{-\pi}^{\pi} |f(x)|^2 dx$

Parseval's Theorem

Parseval's theorem is a generalization of the Pythagorean theorem to the space of square-integrable functions. For Fourier series, it states that the integral of the square of a function over an interval is equal to $\pi$ times the sum of the squares of its Fourier coefficients (with a special term for the constant coefficient). This theorem is incredibly powerful because it equates an integral involving the function itself with a sum involving its Fourier coefficients. This means we can calculate the "energy" of a function by looking at its frequency components, or conversely, we can determine information about the function from its coefficients.

Statement of Parseval's Theorem

Let $f(x)$ be a square-integrable function on $[-\pi, \pi]$ with Fourier coefficients $a_n$ and $b_n$ as defined previously. Parseval's theorem states that:

$\frac{1}{\pi} \int_{-\pi}^{\pi} |f(x)|^2 dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)$

This equation is often referred to as the "energy theorem" for Fourier series.

Significance of Parseval's Theorem

Parseval's theorem provides a direct link between the function and its Fourier series representation in terms of energy. It implies that the total energy of a signal (represented by the integral of its square) is precisely equal to the sum of the energies of its individual frequency components. This is fundamental in signal processing, where it allows for the analysis of signal power distribution across different frequencies. It also confirms that if the Fourier series converges in the mean square sense (as guaranteed by the Riesz–Fischer theorem), then Bessel's inequality becomes an equality.

Proof Idea (Conceptual)

The proof of Parseval's theorem builds upon the previous concepts. We start again with the integral of the square of the difference between $f(x)$ and its partial sum $S_N(x)$:

$\int_{-\pi}^{\pi} |f(x) - S_N(x)|^2 dx = \int_{-\pi}^{\pi} |f(x)|^2 dx - \pi \left( \frac{a_0^2}{2} + \sum_{n=1}^{N} (a_n^2 + b_n^2) \right)$

The Riesz–Fischer theorem guarantees that $\lim_{N \to \infty} \int_{-\pi}^{\pi} |f(x) - S_N(x)|^2 dx = 0$.

Taking the limit as $N \to \infty$ on both sides of the equation:

$0 = \lim_{N \to \infty} \left( \int_{-\pi}^{\pi} |f(x)|^2 dx - \pi \left( \frac{a_0^2}{2} + \sum_{n=1}^{N} (a_n^2 + b_n^2) \right) \right)$

Since $\int_{-\pi}^{\pi} |f(x)|^2 dx$ does not depend on $N$, we can write:

$0 = \int_{-\pi}^{\pi} |f(x)|^2 dx - \pi \left( \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2) \right)$

Rearranging this equation gives Parseval's theorem:

$\int_{-\pi}^{\pi} |f(x)|^2 dx = \pi \left( \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2) \right)$

Or, dividing by $\pi$:

$\frac{1}{\pi} \int_{-\pi}^{\pi} |f(x)|^2 dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)$

Example Application of Parseval's Theorem

Consider the function $f(x) = x$ on the interval $[-\pi, \pi]$. This function is odd, so $a_n = 0$ for all $n \ge 0$.

The coefficients $b_n$ are calculated as:

$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x \sin(nx) dx$

Using integration by parts, we find $b_n = \frac{2(-1)^{n+1}}{n}$ for $n \ge 1$.

Now, let's apply Parseval's theorem:

The left side:

$\frac{1}{\pi} \int_{-\pi}^{\pi} x^2 dx = \frac{1}{\pi} \left[ \frac{x^3}{3} \right]_{-\pi}^{\pi} = \frac{1}{\pi} \left( \frac{\pi^3}{3} - \frac{(-\pi)^3}{3} \right) = \frac{1}{\pi} \left( \frac{\pi^3}{3} + \frac{\pi^3}{3} \right) = \frac{2\pi^2}{3}$

The right side:

$\frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2) = 0 + \sum_{n=1}^{\infty} \left( 0^2 + \left(\frac{2(-1)^{n+1}}{n}\right)^2 \right) = \sum_{n=1}^{\infty} \frac{4}{n^2}$

Equating the two sides according to Parseval's theorem:

$\frac{2\pi^2}{3} = \sum_{n=1}^{\infty} \frac{4}{n^2}$

Dividing by 4, we get a famous result:

$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$

This example demonstrates how Parseval's theorem can be used to find the sum of infinite series.