Fourier Analysis – Comprehensive Reference Guide

Fourier Series - Trig Form

Periodic function on $[-L,L]$:

$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos\frac{n\pi x}{L} + b_n \sin\frac{n\pi x}{L}\right)$

Coefficients

$a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\frac{n\pi x}{L} dx$

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

$a_0/2$ is the average value of $f$ on $[-L,L]$

Fourier Series - Complex Form

$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{in\pi x/L}$

Complex Coefficients

$c_n = \frac{1}{2L} \int_{-L}^{L} f(x) e^{-in\pi x/L} dx$

Relation to Real

$c_0 = a_0/2$

$c_n = (a_n - ib_n)/2$ for $n > 0$

$c_{-n} = \overline{c_n}$ (conjugate)

Complex form unifies positive & negative frequencies

Coefficient Properties

Symmetry

Even $f$: only $a_n$ nonzero (cosine series)

Odd $f$: only $b_n$ nonzero (sine series)

Decay Rates

Continuous $f$: $a_n, b_n = O(1/n)$

$C^k$ smooth: $a_n, b_n = O(1/n^{k+1})$

Discontinuities: slow $O(1/n)$ decay

Faster decay = smoother function

Riemann-Lebesgue

$\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = 0$

Parseval's Theorem - Series

$\frac{1}{2L}\int_{-L}^{L}|f(x)|^2 dx = \frac{|a_0|^2}{4} + \frac{1}{2}\sum_{n=1}^{\infty}(|a_n|^2 + |b_n|^2)$

Complex Form

$\frac{1}{2L}\int_{-L}^{L}|f(x)|^2 dx = \sum_{n=-\infty}^{\infty}|c_n|^2$

Energy in time = energy in frequency domain

Bessel's Inequality

$\sum |c_n|^2 \leq \frac{1}{2L}\int |f|^2$ (equality iff complete)

Dirichlet Conditions

Sufficient for convergence:

$f$ absolutely integrable on $[-L,L]$

Finite number of extrema

Finite number of discontinuities

Convergence Point

At continuity: $S_N(x) \to f(x)$

At jump: $S_N(x) \to \frac{f(x^+)+f(x^-)}{2}$

Series converges at average of left/right limits

Gibbs Phenomenon

Overshoot at discontinuities

Max overshoot: ~8.9% beyond jump

Persists at all $N$ (doesn't vanish)

Oscillations concentrated near jump

Ringing artifacts in truncated series

Mitigation

Cesàro summation smooths ripples

Lanczos/Fejér kernels reduce overshoot

Windowing in signal processing

Convergence Types

Pointwise

$S_N(x) \to f(x)$ for each $x$

Typical for Fourier series

Uniform

$\|S_N - f\|_\infty \to 0$ everywhere

Requires $f$ continuous & smooth

$L^2$ Convergence

$\|S_N - f\|_2 \to 0$

Weakest: allows jump discontinuities

$L^2$ convergence always holds for periodic $L^2$ functions

Fourier Transform Defined

Standard Form

$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} dx$

Inverse

$f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{2\pi i x \xi} d\xi$

Angular Frequency

$\hat{f}(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx$ where $\omega = 2\pi\xi$

Both forms common; use context consistently

Transform Properties I

Linearity

$\mathcal{F}(af+bg) = a\hat{f} + b\hat{g}$

Translation

$\mathcal{F}(f(x-a)) = e^{-2\pi i a \xi} \hat{f}(\xi)$

Modulation

$\mathcal{F}(e^{2\pi i a x}f(x)) = \hat{f}(\xi - a)$

Scaling

$\mathcal{F}(f(ax)) = \frac{1}{|a|}\hat{f}(\xi/a)$

Compression in time = dilation in frequency

Transform Properties II

Convolution

$\mathcal{F}(f*g) = \hat{f} \cdot \hat{g}$

Differentiation

$\mathcal{F}(f') = 2\pi i \xi \hat{f}(\xi)$

Multiplication

$\mathcal{F}(xf(x)) = \frac{i}{2\pi}\hat{f}'(\xi)$

Moments

$\int x^n f(x) dx = \frac{1}{(2\pi i)^n}\hat{f}^{(n)}(0)$

Differentiation amplifies high frequencies (ill-posed)

Parseval & Plancherel

Plancherel Theorem

$\int_{-\infty}^{\infty}|f(x)|^2 dx = \int_{-\infty}^{\infty}|\hat{f}(\xi)|^2 d\xi$

Parseval's Identity

$\int f(x)\overline{g(x)} dx = \int \hat{f}(\xi)\overline{\hat{g}(\xi)} d\xi$

Autocorrelation

$R(x) = f*\bar{f}$ has $\mathcal{F}(R) = |\hat{f}|^2$

Energy conservation fundamental to signal processing

Common Transform Pairs

$f(x)$	$\hat{f}(\xi)$
$\delta(x)$	$1$
$1$	$\delta(\xi)$
$e^{-\pi x^2}$	$e^{-\pi\xi^2}$
$\text{rect}(x)$	$\text{sinc}(\xi)$
$e^{-a\|x\|}$	$\frac{2a}{a^2+(2\pi\xi)^2}$
$\cos(2\pi\omega_0 x)$	$\frac{1}{2}[\delta(\xi-\omega_0)+\delta(\xi+\omega_0)]$
$e^{-ax^2}$	$\sqrt{\frac{\pi}{a}}e^{-\pi^2\xi^2/a}$

Discrete Fourier Transform

DFT Definition

$X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i kn/N}$, $k=0,\ldots,N-1$

Inverse DFT

$x_n = \frac{1}{N}\sum_{k=0}^{N-1} X_k e^{2\pi i kn/N}$

FFT Algorithm

Cooley-Tukey: $O(N\log N)$ vs $O(N^2)$

Requires $N = 2^m$ typically

Separates even/odd indices recursively

FFT standard for signal processing

Sampling Theory

Nyquist-Shannon Theorem

Sample at $f_s \geq 2f_{\max}$ to recover band-limited signal

Nyquist Frequency

$f_N = f_s/2$ (max frequency recoverable)

Aliasing

Occurs when $f_s < 2f_{\max}$

High freqs appear as low freqs

Aliased freq: $f_a = |f - nf_s|$

Alias artifacts permanent; apply anti-alias filter before sampling

Reconstruction Formula

Whittaker-Shannon

$f(t) = \sum_{n=-\infty}^{\infty} f(nT) \text{sinc}\left(\frac{t-nT}{T}\right)$

where $T=1/f_s$ is sample period

Practical Filters

Ideal: infinite sinc (unrealizable)

Linear interp: piecewise constant

Cubic spline: smoother

Trade-off: smoothness vs causality

Laplace Transform

Definition

$F(s) = \int_0^{\infty} f(t) e^{-st} dt$, $s=\sigma+i\omega$

Connection to FT

Laplace: causal, defines $s$-plane

Fourier: non-causal, imaginary axis

Set $s = i\omega$ in ROC for FT

Applications

Solve ODEs/PDEs with IC

Transfer functions & control

More general; includes exponentially growing functions

Key Applications

Signal Processing

Filtering (LP, HP, BP)

Spectral analysis

Compression (JPEG, MP3)

PDEs

Heat equation

Wave equation

Schrödinger equation

Image Processing

2D Fourier transforms

Edge detection

Deblurring via deconvolution

Property	Fourier Series	Fourier Transform	DFT/FFT
Domain	Periodic on $[-L,L]$	Entire real line	Finite discrete sequence
Frequency	Discrete: $n\pi/L$	Continuous: $\xi \in \mathbb{R}$	Discrete: $k/N$
Result	Coefficients $c_n$	Function $\hat{f}(\xi)$	Vector $X_k$
Computation	Integration (analytic)	Integration (analytic)	FFT $O(N\log N)$
Invertible	Yes (if $L^2$)	Yes (if $L^1$)	Yes
Use Case	Periodic signals, PDEs	Theory, analysis	Numerical computation

FOURIER ANALYSIS Complete Reference