Parseval's theorem: Question bank — Hinglish deep dive

4.7.7 · D5 · Maths › Partial Differential Equations › Parseval's theorem

Error dhundho

Har line mein ek student ka step diya hai. Galti dhundho aur theek karo.

(- π, π)

par maine likha

\int_{- π}^{π} f^{2} d x = \frac{a _{0}^{2}}{2} + \sum (a_{n}^{2} + b_{n}^{2})

Left side par

\frac{1}{L} = \frac{1}{π}

factor missing hai; sahi hai

\frac{1}{π} \int_{- π}^{π} f^{2} d x = \frac{a _{0}^{2}}{2} + \sum (a_{n}^{2} + b_{n}^{2})

"DC term right side mein

a_{0}^{2}

contribute karta hai, doosron ke pattern se match karte hue."

Galat — series mein

\frac{a _{0}}{2}

hota hai, toh square karne par

\frac{a _{0}^{2}}{4}

aata hai, aur

\int 1 d x = 2 L

(doosron se double), mil ke

\frac{a _{0}^{2}}{2}

banta hai,

a_{0}^{2}

nahi.

"Kyunki

x

(- π, π)

par odd hai, iski energy

\frac{1}{π} \int x^{2} d x

zero hai."

f

aur

f^{2}

mein confusion ho gayi;

x

odd hai lekin

x^{2}

even aur positive hai, toh uski energy genuinely nonzero hai (

= \frac{2 π ^{2}}{3}

"Mujhe

b_{n} = \frac{2 ( - 1 ) ^{n + 1}}{n}

mila, toh right side hai

\sum b_{n}

Parseval squares ka sum karta hai:

\sum b_{n}^{2} = \sum \frac{4}{n ^{2}}

, na ki

\sum b_{n}

(jo na theek se converge karta aur na energy represent karta).

"Ek hi

f

ki half-range sine series aur full-range series ka Parseval sum same hona chahiye."

Zaroori nahi — ye alag-alag intervals par expansions hain jo

f

ko alag represent karte hain, toh dono energies aur coefficients alag honge; Parseval sirf ek consistent expansion ke andar hold karta hai.

\int_{- L}^{L} cos \frac{mπ x}{L} sin \frac{nπ x}{L} d x = 0

sirf tab hota hai jab

m \neq = n

Ye sabhi

m, n

ke liye

0

hai (including

m = n

), kyunki cosine times sine ek symmetric interval par odd function banta hai.

Kyun wale sawaal

Sirf fact nahi, deep reason batao.

Orthogonality cross terms ko merely cancel karne ki jagah kyun vanish kara deta hai?

Kyunki har cross-term integral individually zero hota hai — perpendicular basis functions ka "overlap" zero hota hai, jaise

e_{x} \cdot e_{y} = 0

, toh kuch bhi cancel karne ki zaroorat nahi.

Engineering mein Parseval ko energy ya power statement kyun kaha jaata hai?

\int f^{2}

signal ki total energy measure karta hai, aur

\frac{a _{n}^{2} + b _{n}^{2}}{2}

n

-th harmonic ki power hai, toh theorem literally kehta hai total power = harmonic powers ka sum (RMS and Power Spectra).

Derivation ki final line ko

L

se divide karne se formula "nicer" kyun lagta hai?

\int f^{2}

ko ek mean value mein convert karta hai aur har survivor ko

\frac{1}{L}

se rescale karta hai, toh

a_{n}^{2}

b_{n}^{2}

terms se

L

hat jaata hai aur DC term exactly

\frac{a _{0}^{2}}{2}

pattern par land karta hai.

Parseval

\sum 1/ n^{2}

jaise impossible-lagte infinite sums kyun evaluate kar sakta hai?

Ye ek known integral (easy) ko ek sum built from coefficients (unknown series) ke barabar rakhta hai, jisse tum sum solve kar sako — summation problem ko integration problem mein badal deta hai (Basel Problem).

Theorem ko basis ke complete hone ki zaroorat kyun hai, sirf orthogonal hone ki nahi?

Orthogonality akela Bessel ka

\leq

deta hai; sirf completeness guarantee karta hai ki koi energy basis ke bahar kisi direction mein nahi jaati, jisse

\leq

exact

=

mein upgrade hota hai.

f (x) = x^{2}

ki Parseval identity

f (x) = x

se "deeper" kyun hai?

Extra power sum ko

\sum 1/ n^{2}

\sum 1/ n^{4} = \frac{π ^{4}}{90}

tak le jaati hai;

x

ki higher powers higher zeta values tak pahunchti hain, results ki poori ek ladder reveal karti hain.

Edge cases

Wo scenarios jo log kabhi test nahi karte — lekin ya toh naive formulas tod dete hain ya structure reveal karte hain.

f (x) = 0

(zero function) ke liye Parseval kya deta hai?

Saare coefficients zero hain aur

\int 0^{2} = 0

, toh dono sides

0

hain — trivial lekin consistent boundary case.

Ek single pure tone

f (x) = b_{1} sin \frac{π x}{L}

ke liye Parseval kya kehta hai?

\frac{1}{L} \int f^{2} = b_{1}^{2}

jab baaki sab coefficients zero hain — energy poori tarah us ek harmonic mein hai, dikhata hai ki har term ek self-contained energy bin hai.

Agar

f

mein jump discontinuity hai, toh kya Parseval phir bhi hold karta hai?

Haan — energy ek integral hai (isolated points ke liye insensitive), toh finite jumps kuch nahi badlate; sirf jump par pointwise value ambiguous hai, total energy nahi.

Agar

f

ke saare coefficients double kar do (yaani

2 f

use karo) toh Parseval ka kya hoga?

Right side

4

se scale hogi (har square ko

2^{2}

factor milega) aur left bhi (

\int (2 f)^{2} = 4 \int f^{2}

) — energy quadratic hai, toh dono sides balanced rehti hain.

Kya right-hand sum finite ho sakta hai jabki

f

ke infinitely many nonzero coefficients hon?

Haan — jab tak

\sum (a_{n}^{2} + b_{n}^{2})

converge kare (jo kisi bhi finite-energy

f

ke liye hona chahiye), coefficients itni fast shrink karte hain ki total finite rahta hai.

Kya Parseval ek aisi function ke liye kaam karta hai jo sirf

(- L, L)

ke ek hisse par nonzero ho, jaise ek pulse?

Haan — integral automatically wahan ignore karta hai jahan

f = 0

, aur coefficients pulse ko encode karte hain; pulse ki energy = coefficient-square sum, koi special treatment nahi chahiye.

Non-periodic

f

ke liye Parseval ka continuous-signal analogue kya hai?

Plancherel Theorem:

\int ∣ f ∣^{2} d x = \int ∣ \hat{f} ∣^{2} d ω

— same energy-conservation idea jisme discrete coefficient sum ki jagah Fourier transform hota hai.

Agar signal mein

a_{0} = 0

ho, toh kya iska matlab hai ki wo koi energy carry nahi karta?

Nahi —

a_{0} = 0

ka sirf matlab hai zero average (koi DC offset nahi); energy phir bhi large ho sakti hai, poori tarah AC harmonics

\sum (a_{n}^{2} + b_{n}^{2})

mein stored.

Connections

Parseval's theorem — wo parent statement jise ye traps stress-test karte hain.
Orthogonality of functions — kyun cross terms exactly zero hote hain.
Bessel's Inequality — woh $\leq$ jise Parseval $=$ mein sharpen karta hai.
Fourier Series — wo expansion jis par ye poora theorem based hai.
Basel Problem — famous payoff sum.
Plancherel Theorem — transform-world ka twin.
RMS and Power Spectra — "energy" ka engineering reading.

Question bank — Parseval's theorem

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