Fourier Methods

1. Fourier Series: Derivation from Orthogonality

Consider a function $f(x)$ defined on the interval $[-L, L]$ (or equivalently, a periodic function with period $2L$). We wish to expand it in the basis of trigonometric functions:

1.1 Orthogonality Relations

The key to determining the coefficients is the orthogonality of the trigonometric functions on $[-L, L]$. We derive these from scratch:

Cosine-cosine: For integers $n, m \ge 0$:

Proof: Using the product-to-sum identity$\cos\alpha\cos\beta = \frac{1}{2}[\cos(\alpha-\beta) + \cos(\alpha+\beta)]$:

For $n \neq m$, each integral is $\frac{L}{k\pi}\sin(k\pi)|_{-L}^{L} = 0$ since$\sin(k\pi) = 0$ for integer $k$. For $n = m \neq 0$, the first integral gives$2L$ and the second gives $0$, yielding $L$.

Similarly, $\sin$-$\sin$ orthogonality gives $L\delta_{nm}$, and$\sin$-$\cos$ integrals always vanish (the integrand is odd).

1.2 Deriving the Coefficients

To find $a_n$, multiply the Fourier series by $\cos(m\pi x/L)$ and integrate over $[-L, L]$. By orthogonality, every term vanishes except the one with $n = m$:

Therefore:

The factor of $1/2$ in front of $a_0$ is a convention that makes the formula for $a_0$ consistent with the general formula (since the norm of the constant function is$2L$ rather than $L$).

1.3 Complex Fourier Series

Using $e^{in\pi x/L} = \cos(n\pi x/L) + i\sin(n\pi x/L)$, the Fourier series can be written more compactly:

The orthogonality relation in complex form is:

2. The Fourier Transform

The Fourier transform extends Fourier series to non-periodic functions by taking $L \to \infty$. As the period grows, the discrete frequencies $n\pi/L$ become continuous, the sum becomes an integral, and the coefficients become a continuous function.

2.1 Derivation from Fourier Series

Starting from the complex Fourier series with $k_n = n\pi/L$ and $\Delta k = \pi/L$:

In the limit $L \to \infty$, $\Delta k \to dk$ and the sum becomes an integral:

Defining the Fourier transform and its inverse (physicist's convention):

2.2 Important Transform Pairs

Gaussian: The Fourier transform of a Gaussian is a Gaussian:

Proof: Complete the square in the exponent:

This result embodies the uncertainty principle: narrowing the Gaussian in $x$-space (large $a$) broadens it in $k$-space, and vice versa. The product of widths$\Delta x \cdot \Delta k \ge 1/2$ is minimized for Gaussians.

Rectangular pulse:

3. Parseval's Theorem

Parseval's theorem (also called Plancherel's theorem in the transform context) states that the total energy is the same in both real and frequency space.

3.1 For Fourier Series

Theorem:

Proof: Using the complex form:

3.2 For Fourier Transforms

Physical interpretation: In quantum mechanics, $|\psi(x)|^2$ is the position-space probability density and $|\tilde{\psi}(k)|^2/(2\pi)$ is the momentum-space probability density. Parseval's theorem ensures total probability is conserved regardless of which representation we use.

4. The Convolution Theorem

The convolution of two functions is defined as:

Convolution theorem: Convolution in real space corresponds to multiplication in Fourier space:

Proof:

Substituting $y = x - x'$ (so $x = y + x'$):

Application: This theorem is the foundation of signal processing. A linear time-invariant system with impulse response $h(t)$ transforms an input $f(t)$into output $y(t) = f * h$. In Fourier space, $\tilde{y} = \tilde{f}\,\tilde{h}$, making filter design simply a matter of choosing $\tilde{h}(k)$.

There is also a dual result: multiplication in real space corresponds to convolution in Fourier space (divided by $2\pi$):

5. The Dirac Delta Function

The Dirac delta $\delta(x)$ is not a function in the classical sense but a distribution (generalized function) defined by its action:

5.1 Fourier Representation

The Fourier transform of $\delta(x)$ is:

Therefore, the inverse transform gives the crucial Fourier representation of the delta function:

This identity is fundamental throughout physics. It expresses the completeness of the plane wave basis $\{e^{ikx}\}$ and is the continuous analog of the Kronecker delta.

5.2 Representations as Limits

The delta function can be represented as a limit of ordinary functions (nascent deltas):

• Gaussian: $\delta(x) = \lim_{\epsilon \to 0} \frac{1}{\sqrt{2\pi}\epsilon}\,e^{-x^2/(2\epsilon^2)}$
• Lorentzian: $\delta(x) = \lim_{\epsilon \to 0} \frac{1}{\pi}\frac{\epsilon}{x^2 + \epsilon^2}$
• Sinc: $\delta(x) = \lim_{N \to \infty} \frac{\sin(Nx)}{\pi x}$

5.3 Properties

• Scaling: $\delta(ax) = \frac{1}{|a|}\delta(x)$
• Composition: $\delta(g(x)) = \sum_i \frac{\delta(x - x_i)}{|g'(x_i)|}$ where $x_i$ are the zeros of $g$
• Derivative: $\int f(x)\,\delta'(x-a)\,dx = -f'(a)$ (integration by parts)
• Sifting: $f(x)\,\delta(x-a) = f(a)\,\delta(x-a)$

6. Application: Heat Equation on an Infinite Domain

Consider the heat equation on the entire real line:

Step 1: Fourier transform in $x$. Let $\tilde{u}(k,t) = \int u(x,t)\,e^{-ikx}\,dx$:

Step 2: By the convolution theorem, since $\tilde{u} = \tilde{f} \cdot \tilde{G}$where $\tilde{G}(k,t) = e^{-\alpha k^2 t}$, we have $u = f * G$. The inverse transform of $\tilde{G}$ is:

This is the heat kernel (or Green's function for the heat equation). The solution is:

This shows that heat diffusion is a Gaussian convolution: each initial point source spreads as a Gaussian whose width grows as $\sqrt{\alpha t}$. Sharp features are smoothed out, reflecting the irreversible nature of diffusion.

7. Applications to Signal Processing

Fourier methods are the backbone of modern signal processing. Key concepts include:

7.1 The Sampling Theorem (Nyquist-Shannon)

A band-limited signal with maximum frequency $f_{\max}$ can be perfectly reconstructed from samples taken at rate $f_s \ge 2f_{\max}$ (the Nyquist rate). Sampling at a lower rate causes aliasing, where high-frequency components masquerade as low frequencies.

7.2 The Discrete Fourier Transform

For $N$ equally spaced samples $\{f_j\}_{j=0}^{N-1}$, the DFT is:

The FFT algorithm computes this in $O(N\log N)$ operations instead of $O(N^2)$, making spectral analysis practical for large datasets.

7.3 Windowing and Spectral Leakage

Truncating a signal is equivalent to multiplying by a rectangular window. By the convolution theorem, this convolves the spectrum with a sinc function, causing spectral leakage. Smooth window functions (Hann, Hamming, Blackman) reduce leakage at the cost of frequency resolution.

8. Multi-Dimensional Fourier Transforms

The Fourier transform generalizes naturally to $d$ dimensions:

Radial symmetry: If $f(\mathbf{r})$ depends only on $r = |\mathbf{r}|$, the angular integration can be performed analytically. In 3D:

This is the Hankel transform of order zero (up to factors). It appears frequently in scattering theory and the analysis of radially symmetric potentials.

Application to Coulomb potential: The Fourier transform of $V(r) = e^2/(4\pi\epsilon_0 r)$in 3D gives:

This is the basis of the momentum-space treatment of the Coulomb interaction in quantum field theory and condensed matter physics. The $1/k^2$ form directly reflects the $1/r$potential via the relation $\nabla^2(1/r) = -4\pi\delta^3(\mathbf{r})$ and the Fourier representation of the Laplacian: $\nabla^2 \to -k^2$.

9. Gibbs Phenomenon and Convergence

When a Fourier series is used to represent a function with jump discontinuities, the partial sums exhibit characteristic overshoots near the discontinuity that do not diminish as more terms are added. This is the Gibbs phenomenon.

For a step function, the overshoot is approximately $8.95\%$ of the jump magnitude, regardless of the number of terms. Mathematically, this arises because:

where $x_N$ is the location of the first maximum near the discontinuity, which approaches the discontinuity as $N \to \infty$.

Types of convergence:

• Pointwise: $S_N(x) \to f(x)$ at each continuity point. At jumps, converges to the average $\frac{1}{2}[f(x^+) + f(x^-)]$.
• Uniform: $\|S_N - f\|_\infty \to 0$. Requires continuity. Fails for discontinuous functions.
• Mean-square (L2): $\|S_N - f\|_2 \to 0$. Guaranteed for all square-integrable functions by completeness.

The rate of convergence depends on the smoothness of $f$: if $f \in C^k$ (k times continuously differentiable), the Fourier coefficients decay as $O(1/n^{k+1})$. Infinitely smooth periodic functions have exponentially decaying coefficients.

Cesaro summation provides a way to eliminate the Gibbs phenomenon. The Cesaro (or Fejer) partial sums are the arithmetic means of the ordinary partial sums:

where $F_N(t) = \frac{1}{N+1}\frac{\sin^2((N+1)t/2)}{\sin^2(t/2)}$ is the Fejer kernel. Unlike the Dirichlet kernel, the Fejer kernel is non-negative, which guarantees that $\sigma_N$converges uniformly to $f$ for continuous $f$, without any Gibbs overshoot.

8. Python Simulation: Fourier Methods

This simulation verifies Fourier coefficient formulas, Parseval's theorem, the convolution theorem, and demonstrates the heat kernel solution.