Fourier Reciprocity & Parseval's Identity

Energy Conservation and Convergence on the Circle

We begin by considering the Fourier Reciprocity Formula and how it naturally extends to Parseval's Identity. Consider a function represented by its complex Fourier series:

f(\theta) = \sum_{n=-\infty}^{\infty}C_ne^{in\theta}, \quad C_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(\theta)e^{-in\theta}d\theta

Convergence and the Weierstrass M-Test

To ensure our function is continuous and bounded, we assume the coefficients are absolutely summable: \(\sum |C_n| < \infty\). By the Weierstrass M-Test, this guarantees uniform and absolute convergence.

The M-Test: If \(|f_n(x)| \leq M_n\) and \(\sum M_n\) converges, then \(\sum f_n(x)\) converges uniformly.

In our case: \(|C_n e^{in\theta}| = |C_n|\). Since \(\sum |C_n|\) converges, our function \(f(\theta)\) is continuous and finite on the circle \(\mathbb{T} = (-\pi, \pi]\).

The Orthogonality Principle

The backbone of this proof is the orthogonality of the complex exponentials:

\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-im\theta} e^{in\theta} d\theta = \begin{cases} 1 & \text{if } n = m \\ 0 & \text{if } n \neq m \end{cases}

Deriving the Reciprocity Formula

Let \(g \in L^1(\mathbb{T})\) with Fourier coefficients \(D_n\). We look at the integral of the product of two functions:

\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\theta)g(\theta)d\theta = \frac{1}{2\pi}\sum_{n \in \mathbb{Z}}C_n\left(\int_{-\pi}^{\pi}g(\theta)e^{in\theta}d\theta\right)

Note that \(\int g(\theta)e^{in\theta}d\theta\) is almost the definition of \(D_{-n}\). This gives us:

Fourier Reciprocity Formula

\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\theta)g(\theta)d\theta = \sum_{n\in\mathbb{Z}}C_nD_{-n}

Parseval's Identity

By letting \(g = \bar{f}\) (the complex conjugate), we transform the left side into the energy of the signal. Since \(f\bar{f} = |f|^2\), we ensure the result is real-valued.

Parseval's Identity on the Circle

\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|f(\theta)\right|^2d\theta = \sum_{n \in \mathbb{Z}}\left|C_n\right|^2

This identity is profound: it states that the total "energy" of a signal (left) is equal to the sum of the energies of its individual frequency components (right).

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