Fourier transform — definition, properties

1. Deriving the transform from Fourier series (from scratch)

A function periodic with period $2L$ has the complex Fourier series $f(x)=\sum_{n=-\infty}^{\infty} c_n\, e^{i k_n x},\qquad k_n=\frac{n\pi}{L},\qquad c_n=\frac{1}{2L}\int_{-L}^{L} f(x)\,e^{-ik_n x}\,dx.$

Why this form? The waves $e^{ik_n x}$ are orthogonal over one period, so $c_n$ is just the "projection" of $f$ onto that wave.

Now let $L\to\infty$ . The spacing between frequencies is $\Delta k=k_{n+1}-k_n=\frac{\pi}{L}\to 0$ . Define $\hat f(k)=\int_{-\infty}^{\infty} f(x)\,e^{-ikx}\,dx \quad(\text{the } 2L\,c_n \text{ in the limit}).$

Substitute $c_n=\frac{\Delta k}{2\pi}\hat f(k_n)$ into the series: $f(x)=\sum_n \frac{\Delta k}{2\pi}\,\hat f(k_n)\,e^{ik_n x}\;\xrightarrow{\Delta k\to 0}\;\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat f(k)\,e^{ikx}\,dk.$

The Riemann sum became an integral. That gives us the pair:

Recall Why is the inverse the "same" integral with

$+ik$ and a $1/2\pi$ ? Because the waves are orthogonal: $\int_{-\infty}^{\infty}e^{i(k-k')x}\,dx=2\pi\,\delta(k-k')$ . Plugging $\hat f$ into the inverse and using this delta collapses the double integral back to $f(x)$ . The $2\pi$ from the delta is what the $1/2\pi$ cancels.

2. Core properties (each derived)

Figure — Fourier transform — definition, properties

3. Worked examples

Worked example (1) The Gaussian — its own transform

Compute $\mathcal F\{e^{-x^2/2}\}$ . $\hat f(k)=\int_{-\infty}^{\infty}e^{-x^2/2}e^{-ikx}dx.$ Step — complete the square (Why? to turn the exponent into a perfect Gaussian we can integrate): $-\tfrac{x^2}{2}-ikx=-\tfrac12(x+ik)^2-\tfrac{k^2}{2}.$ So $\hat f(k)=e^{-k^2/2}\int e^{-\frac12(x+ik)^2}dx=\sqrt{2\pi}\,e^{-k^2/2}.$ Why the last step? Shifting the contour by $ik$ leaves the standard integral $\int e^{-u^2/2}du=\sqrt{2\pi}$ . Lesson: the Gaussian is an eigenfunction of $\mathcal F$ . Narrow Gaussian ( $x^2/2\sigma^2$ , small $\sigma$ ) ⇒ wide transform — uncertainty in action.

Worked example (3) Solving the heat equation with the transform

Solve $u_t=\alpha u_{xx}$ , $u(x,0)=f(x)$ , on $-\infty<x<\infty$ . Step 1 — transform in $x$ : let $\hat u(k,t)=\mathcal F\{u\}$ . Using $\mathcal F\{u_{xx}\}=(ik)^2\hat u=-k^2\hat u$ : $\hat u_t=-\alpha k^2\,\hat u.$ Why? The PDE became an ODE in $t$ for each fixed $k$ . Step 2 — solve the ODE: $\hat u(k,t)=\hat f(k)e^{-\alpha k^2 t}$ . Step 3 — invert: product in $k$ ⇒ convolution in $x$ . Since $\mathcal F^{-1}\{e^{-\alpha k^2 t}\}=\frac{1}{\sqrt{4\pi\alpha t}}e^{-x^2/4\alpha t}$ , $u(x,t)=\frac{1}{\sqrt{4\pi\alpha t}}\int_{-\infty}^{\infty}f(\xi)\,e^{-(x-\xi)^2/4\alpha t}\,d\xi.$ The Gaussian heat kernel emerges automatically. This is WHY the transform is in your PDE course.

4. Common mistakes (steel-manned)

5. Flashcards

What is the forward Fourier transform (our convention)?

\hat f(k)=\int_{-\infty}^{\infty} f(x)e^{-ikx}dx

What is the inverse Fourier transform?

f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat f(k)e^{ikx}dk

\mathcal F\{f'(x)\}=?

ik\,\hat f(k)

(assuming

f\to0

\pm\infty

)

\mathcal F\{f^{(n)}\}=?

(ik)^n\hat f(k)

\mathcal F\{f(x-a)\}=?

e^{-ika}\hat f(k)

(phase shift, magnitude unchanged)

\mathcal F\{e^{ik_0x}f(x)\}=?

\hat f(k-k_0)

(spectral shift)

\mathcal F\{f(ax)\}=?

\frac{1}{|a|}\hat f(k/a)

Convolution theorem:

\mathcal F\{f*g\}=?

\hat f(k)\hat g(k)

Parseval's theorem?

\int|f|^2dx=\frac{1}{2\pi}\int|\hat f|^2dk

Fourier transform of

e^{-x^2/2}

\sqrt{2\pi}\,e^{-k^2/2}

(Gaussian is an eigenfunction)

Fourier transform of a box of half-width

a

\frac{2\sin(ka)}{k}=2a\,\mathrm{sinc}(ka)

Why is FT useful in PDEs?

It converts

\frac{d}{dx}

into

\times ik

, turning a PDE into an algebraic/ODE problem

What is

\int_{-\infty}^\infty e^{i(k-k')x}dx

2\pi\,\delta(k-k')

(orthogonality of waves)

Recall Feynman: explain to a 12-year-old

Imagine a song. Your ear hears it as a mess of pressure over time — that's $f(x)$ . But a music app shows you bars: how much bass, how much treble. That picture of "how much of each pitch" is the Fourier transform $\hat f(k)$ . The song and the bar chart hold the same information, just shown two ways. The cool trick: some problems that look horrible as a song become super easy as a bar chart — like figuring out how a hot bar of metal cools down. You switch to the bar chart, do easy arithmetic on each bar, then switch back. That switching machine is the Fourier transform.

Connections

Fourier Series — the discrete/periodic ancestor; FT is its $L\to\infty$ limit.
Heat Equation — solved cleanly via FT (worked example 3).
Convolution — the operation FT turns into multiplication.
Dirac Delta Function — provides the orthogonality $\int e^{i(k-k')x}dx=2\pi\delta$ .
Laplace Transform — cousin transform for half-line / initial-value problems.
Uncertainty Principle — narrow in $x$ ⟺ wide in $k$ (scaling property).
Separation of Variables — alternative PDE method; FT replaces it on infinite domains.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Fourier transform ka basic idea simple hai: koi bhi function $f(x)$ ko hum alag-alag frequencies ke waves ka mixture maan sakte hain. $\hat f(k)$ basically batata hai ki frequency $k$ ki kitni "matra" us function mein hai. Jaise gaane mein bass aur treble kitna hai — wahi cheez maths mein $\hat f(k)$ hai. Fourier series periodic function ke liye discrete frequencies deti hai, aur jab period ko infinite kar do, toh sum integral ban jaata hai — yahi se Fourier transform nikalta hai.

PDEs mein iska sabse bada faayda yeh hai: derivative $\frac{d}{dx}$ transform lene par sirf $ik$ se multiply ho jaata hai. Matlab ek mushkil differential equation, frequency space mein ek simple algebra/ODE ban jaati hai. Jaise heat equation $u_t=\alpha u_{xx}$ — transform lo, har $k$ ke liye chhoti si ODE solve karo ( $\hat u=\hat f\,e^{-\alpha k^2 t}$ ), phir wapas inverse transform lo, aur Gaussian heat kernel khud aa jaata hai. Magic!

Properties yaad rakhne layak hain: shift in $x$ se sirf phase $e^{-ika}$ aata hai (magnitude same), scaling mein $x$ squeeze karoge toh $k$ stretch hoga (yahi uncertainty principle hai), aur convolution real space mein = simple multiplication frequency space mein. Gaussian apna khud ka transform hai (eigenfunction), aur box function ka transform sinc banta hai — sharp edge matlab high frequencies zaroori.

Sabse common galti: log $ik$ ke jagah sirf $k$ likh dete hain — $i$ mat bhoolna, warna heat equation decay ke jagah grow karne lagegi. Aur convolution theorem mein convolution → multiply hota hai, convolution → convolution nahi. Bas convention (yeh $1/2\pi$ inverse mein) shuru mein clear kar lo aur consistent raho.

Test yourself — Partial Differential Equations

Connections