Visual walkthrough — Parseval's theorem

4.7.7 · D2 · Maths › Partial Differential Equations › Parseval's theorem

Hum ek waqt mein ek hi idea build karte hain. Pehle figure padho, phir words.

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Step 1 — Kisi function ki "energy" kya hoti hai?

Square pehle kyun karte hain? Agar hum sirf $f$ integrate karte, toh axis ke upar aur neeche ke parts cancel ho jaate aur ek badi wobbly wave zero score kar sakti thi. Squaring se har wiggle ek positive contribution banta hai — bilkul waise jaise ek loud sound wave power carry karta hai chahe pressure normal se upar ho ya neeche.

PICTURE: red curve $f$ hai; grey region $[f(x){]}^2$ hai. Energy grey area hai.

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Step 2 — Function secretly pure tones ka sum hai

Isko split kyun karte hain? Ek messy curve ke baare mein reason karna mushkil hai. Pure sines aur cosines simple hote hain aur — sabse important baat — yeh aapas mein interfere nahi karte (Step 4). Dekhein Fourier Series ki knobs $a_n,b_n$ kaise milti hain.

PICTURE: upar red target curve; neeche teen pure tones jo mila ke woh curve banate hain. Knobs ghuma lo, tones stack karo, curve recover karo.

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Step 3 — EK pure tone ki energy

${\cos }^2$ ka average $\frac{1}{2}$ kyun hai? Figure dekho: ${\cos }^2$ $0$ aur $1$ ke beech bob karta hai, line $y=\frac{1}{2}$ ke baare mein symmetric hai. Us midline ke upar aur neeche ka area match karta hai — toh average height exactly $\frac{1}{2}$ hai. ${\sin }^2$ ke liye bhi yahi story hai.

PICTURE: red curve ${\cos }^2\frac{\pi x}{L}$; dashed line $\frac{1}{2}$ par uska average hai; grey area $\frac{1}{2}$ height ke rectangle ke area ke barabar hai.

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Step 4 — Do alag tones overlap nahi karte (orthogonality)

Yeh matter kyun karta hai? Jab hum poori series ko square karte hain (agla step), hume har pair of tones ke products milte hain. Yeh law kehta hai ki har "mixed" product silently delete ho jaata hai — sirf ek tone times khud survive karta hai. Yahi poora trick hai. Yeh exactly Orthogonality of functions hai — do waves "perpendicular" hain agar unka product integrate karke zero aaye, bilkul waise jaise do arrows perpendicular hoti hain jab unka dot product zero ho.

PICTURE: red $\cos \frac{\pi x}{L}\cos \frac{2\pi x}{L}$ ka product hai. Blue-shaded positive lobes aur grey negative lobes ka identical area hai — woh kuch nahi banate, cancel ho jaate hain.

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Step 5 — Poori series ko square karo aur integrate karo

Diagonal aur cross mein split kyun karte hain? Kyunki Step 4 ne already har family ki fate bata di hai: har cross term integrate karke $0$ deta hai (orthogonality), har diagonal term integrate karke ek clean $L$ deta hai (Step 3). Hum bas table sweep karte hain.

PICTURE: ek grid. Rows aur columns tones hain; har cell ek product hai. Red diagonal cells survive karte hain; har off-diagonal (cross) cell cross-out hai — integration par woh vanish ho jaata hai.

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Step 6 — Survivors ko add karo

Dono sides ko $L$ se divide karo: $\boxed{\frac{1}{L}{\int }_{-L}^L[f(x){]}^{2}dx=\frac{a_0^2}{2}+\sum _{n=1}^{\infty }(a_n^2+b_n^2)}$

$L$ se divide kyun karte hain? Yeh raw energy ko ek average (mean-square) mein convert karta hai aur DC term ko $\frac{a_0^2}{2}$ ke roop mein line up karta hai — baaki sab jaisa hi pattern, bas "aadha coat pehne hue." Yeh Parseval's theorem hai, Bessel's Inequality ka equality case.

PICTURE: left column $f$ ki single grey energy bar hai; right column wahi total height hai, stacked colored bins mein slice ki gayi — har harmonic ke liye ek. Same total, redistribute kiya gaya.

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Step 7 — Degenerate & edge cases (taaki kuch surprise na kare)

Yeh list kyun banate hain? Jo theorem aap stress-test nahi kar sakte, usp pe aap trust nahi karte. Har case mein dono sides match karti hain, aur "missing" coefficients simply khali energy bins matlab rakhte hain, broken formula nahi.

PICTURE: side by side do bar charts — odd function sirf sine bins fill karta hai (red), even function sirf cosine bins fill karta hai. Complementary, kabhi contradictory nahi.

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Ek picture mein poori summary

Sab kuch compressed: wave, uska energy area, uska tones mein decomposition, aur energy-bin bar chart jo same total tak sum honi chahiye. Ise left se right padho aur aapne Parseval re-derive kar liya.

Recall Feynman retelling — poora walkthrough plain words mein

Ek wiggly wave lo. Uski "energy" woh shaded area hai jo use square karne ke baad milta hai (squaring isliye ki kuch cancel na ho). Ab yahan secret hai: woh wave actually ek chord hai — pure tones (sines aur cosines) ka stack, har ek kisi volume knob se upar kiya gaya. Jab aap poore chord ko square karte ho toh har tone times har doosri tone milti hai. Lekin do alag tones, multiply hokar totaled, hamesha cancel hokar kuch nahi dete — woh perpendicular hain, jaise right angles par arrows. Toh saare messy mixed terms vanish ho jaate hain. Sirf har tone-times-itself bachta hai, aur ek single tone ki energy sirf uska volume-knob squared hai (times ek fixed factor). Unhe add karo, average banane ke liye interval width se divide karo, aur aap yahan pahunchte ho: total average energy = squared volume knobs ka sum. Kuch leak nahi hota. Isliye yahi trick $1+\frac{1}{4}+\frac{1}{9}+\cdots =\frac{{\pi }^2}{6}$ add kar sakti hai — Basel Problem dekho — aur isliye engineers ise power in each harmonic ke liye use karte hain.

Connections

• Parseval's theorem — parent note (statement + worked sums).
• Fourier Series — jahan se tones aur knobs $a_n,b_n$ aate hain.
• Orthogonality of functions — Step 4, engine.
• Bessel's Inequality — Parseval uska equality case hai.
• Plancherel Theorem — continuous transform ke liye wahi idea.