Fourier transform — definition, properties

4.7.18 · Maths › Partial Differential Equations

1. Fourier series se transform derive karna (scratch se)

Ek function jo period $2 L$ ke saath periodic hai, uski complex Fourier series hoti hai $f (x) = \sum_{n = - \infty}^{\infty} c_{n} e^{i k_{n} x}, k_{n} = \frac{nπ}{L}, c_{n} = \frac{1}{2 L} \int_{- L}^{L} f (x) e^{- i k_{n} x} d x .$

Yeh form kyun? Waves $e^{i k_{n} x}$ ek period pe orthogonal hain, toh $c_{n}$ bas $f$ ka us wave pe "projection" hai.

Ab $L \to \infty$ karte hain. Frequencies ke beech spacing hai $Δ k = k_{n + 1} - k_{n} = \frac{π}{L} \to 0$ . Define karo $\hat{f} (k) = \int_{- \infty}^{\infty} f (x) e^{- ik x} d x (yeh 2 L c_{n} limit mein hai) .$

Series mein $c_{n} = \frac{Δ k}{2 π} \hat{f} (k_{n})$ substitute karo: $f (x) = \sum_{n} \frac{Δ k}{2 π} \hat{f} (k_{n}) e^{i k_{n} x} Δ k \to 0 \frac{1}{2 π} \int_{- \infty}^{\infty} \hat{f} (k) e^{ik x} d k .$

Riemann sum integral ban gayi. Isse hume milta hai pair:

Recall Inverse "same" integral kyun hai

$+ ik$ aur $1/2 π$ ke saath? Kyunki waves orthogonal hain: $\int_{- \infty}^{\infty} e^{i (k - k^{'}) x} d x = 2 π δ (k - k^{'})$ . $\hat{f}$ ko inverse mein plug karo aur is delta ko use karo — double integral collapse hokar $f (x)$ ban jaata hai. Delta se aaya $2 π$ wahi hai jo $1/2 π$ cancel karta hai.

2. Core properties (har ek derive ki gayi)

Figure — Fourier transform — definition, properties

3. Worked examples

4. Common mistakes (steel-manned)

5. Flashcards

Forward Fourier transform kya hai (hamari convention)?

\hat{f} (k) = \int_{- \infty}^{\infty} f (x) e^{- ik x} d x

Inverse Fourier transform kya hai?

f (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} \hat{f} (k) e^{ik x} d k

F {f^{'} (x)} = ?

ik \hat{f} (k)

(assuming

f \to 0

\pm \infty

)

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

(ik)^{n} \hat{f} (k)

F {f (x - a)} = ?

e^{- ik a} \hat{f} (k)

(phase shift, magnitude unchanged)

F {e^{i k_{0} x} f (x)} = ?

\hat{f} (k - k_{0})

(spectral shift)

F {f (a x)} = ?

\frac{1}{∣ a ∣} \hat{f} (k / a)

Convolution theorem:

F {f * g} = ?

\hat{f} (k) \overset{g}{^} (k)

Parseval's theorem?

\int ∣ f ∣^{2} d x = \frac{1}{2 π} \int ∣ \hat{f} ∣^{2} d k

e^{- x^{2} /2}

ka Fourier transform?

2 π e^{- k^{2} /2}

(Gaussian ek eigenfunction hai)

Half-width

a

ke box ka Fourier transform?

\frac{2 s i n ( k a )}{k} = 2 a sinc (k a)

PDEs mein FT kyun useful hai?

Yeh

\frac{d}{d x}

\times ik

mein convert karta hai, PDE ko algebraic/ODE problem bana deta hai

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

kya hai?

2 π δ (k - k^{'})

(waves ki orthogonality)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek gaana socho. Tumhara kaan use pressure ka ek jumble sunta hai time ke saath — woh hai $f (x)$ . Lekin ek music app tumhe bars dikhata hai: kitna bass hai, kitna treble. "Har pitch kitni hai" ka woh picture Fourier transform $\hat{f} (k)$ hai. Gaana aur bar chart same information rakhte hain, bas do alag tarike se dikhate hain. Cool trick yeh hai: kuch problems jo gaane ke roop mein bahut buri lagti hain, bar chart mein super easy ho jaati hain — jaise yeh pata karna ki ek garam lohe ki rod kaise thandi hoti hai. Tum bar chart pe switch karo, har bar pe aasaan arithmetic karo, phir wapas switch karo. Woh switching machine hai Fourier transform.

Connections

Fourier Series — discrete/periodic ancestor; FT iska $L \to \infty$ limit hai.
Heat Equation — FT se cleanly solve hoti hai (worked example 3).
Convolution — woh operation jise FT multiplication mein badal deta hai.
Dirac Delta Function — orthogonality $\int e^{i (k - k^{'}) x} d x = 2 π δ$ provide karta hai.
Laplace Transform — half-line / initial-value problems ke liye cousin transform.
Uncertainty Principle — $x$ mein narrow ⟺ $k$ mein wide (scaling property).
Separation of Variables — alternate PDE method; FT ise infinite domains pe replace karta hai.