Fourier Analysis and Hilbert Spaces

Introduction to Taylor series allowed us to approximate functions using power series. However, power series are local—they accurately represent a function only near a specific center point. Fourier Analysis provides a global alternative by decomposing functions into infinite sums of sinusoidal oscillations (sines and cosines).

The Fourier Series

For a periodic function $f(x)$ with period $T=2L$ that is piecewise smooth, we can represent it as a Fourier Series: $f(x)=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }\left[a_n\cos \left(\frac{n\pi x}{L}\right)+b_n\sin \left(\frac{n\pi x}{L}\right)\right]$

Euler-Fourier Formulas

The coefficients are determined by the orthogonality of the trigonometric functions:

• $a_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\cos \left(\frac{n\pi x}{L}\right)dx$
• $b_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\sin \left(\frac{n\pi x}{L}\right)dx$

Complex Form and Hilbert Spaces

In advanced analysis, we use the complex exponential form, which is more compact: $f(x)={\sum }_{n=-\infty }^{\infty }{c}_n{e}^{i\frac{n\pi x}{L}},c_n=\frac{1}{2L}{\int }_{-L}^{L}f(x)e^{-i\frac{n\pi x}{L}}dx$

The Geometry of $L^2[-L,L]$

Fourier analysis is most naturally understood in the context of the Hilbert Space $L^2$ of square-integrable functions.

• The inner product of two functions $f,g$ is $⟨f,g⟩={\int }_{-L}^{L}f(x)\overline{g(x)}dx$.
• The functions $e^{i\frac{n\pi x}{L}}$ form an orthonormal basis for this space.

Convergence: Dirichlet Conditions

A function $f(x)$ is represented by its Fourier series at a point of continuity if it satisfies the Dirichlet conditions:

1. $f$ is absolutely integrable over a period.
2. $f$ has a finite number of maxima and minima within any finite interval.
3. $f$ has a finite number of discontinuities, none of which are infinite.

At a point $x_0$ where $f$ has a jump discontinuity, the Fourier series converges to the average value: $\frac{1}{2}[f(x_0^{+})+f(x_0^{-})]$.

The Fourier Transform

When the period $L\to \infty$, the discrete sum becomes a continuous integral, known as the Fourier Transform: $\hat{f}(\xi )={\int }_{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }dx$ The inverse transform allows us to reconstruct the function: $f(x)={\int }_{-\infty }^{\infty }\hat{f}(\xi )e^{2\pi ix\xi }d\xi$

Parseval’s Theorem

One of the most profound results in Fourier analysis is the preservation of “energy” between the space and frequency domains: ${\sum }_{n=-\infty }^{\infty }|c_n{|}^2=\frac{1}{2L}{\int }_{-L}^L|f(x){|}^{2}dx$ In the continuous case, this is known as Plancherel’s Theorem: $\Vert \hat{f}{\Vert }_2=\Vert f{\Vert }_2$.

Python: Computing a Discrete Fourier Transform (DFT)

We can use the Fast Fourier Transform (FFT) algorithm to analyze the frequency components of a signal.

import numpy as np

# Create a signal with two frequencies: 50Hz and 120Hz
fs = 1000 # Sampling frequency
t = np.linspace(0, 1, fs)
signal = np.sin(2 * np.pi * 50 * t) + 0.5 * np.sin(2 * np.pi * 120 * t)

# Compute the FFT
fft_vals = np.fft.fft(signal)
frequencies = np.fft.fftfreq(len(t), 1/fs)

# Find the peak frequencies
indices = np.where(np.abs(fft_vals) > 100)
print(f"Detected frequencies: {frequencies[indices][:2]}")

Significance

Fourier analysis is the foundation of signal processing, acoustics, and quantum mechanics (where the position and momentum of a particle are Fourier transform pairs). It allows us to solve partial differential equations (like the Heat Equation and Wave Equation) by transforming them into simpler algebraic equations in the frequency domain.

Exercise