# Questions tagged [parsevals-identity]

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### Question on Parseval's Theorem and Plancherel’s formula

I've come across Parseval's theorem and Plancherel’s formula several times on this forum. Each time they're referenced they're mentioned in regards to inner products in general. However, every proof I ...
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### Proving parseval identity for trigonometric polynomials

Show that $$||P||_2^2=\sum_{k=-N}^N \langle P,e_k\rangle^2,$$ where $e_k$ are the Fourier basic functions, and $P$ is a trigonometric polynomial of degree $N.$ I am not sure how to link trigonometric ...
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### Prove a relationship established between the norm of a norm of a function and that of its derivative

From the book Fourier Analysis an Introduction Chapter 3 Exercise 11 b,c b) If $f$ is $T$-periodic, continuous, and piecewise $C^{1}$ with $\int_{0}^{T}f(t)dt=0,$ and $g$ is just $C^{1}$ and $T$-...
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### Hypotheses on Plancherel's theorem

Plancherel's theorem is stated as (e.g. in Rudin's Real and Complex Analysis) If $f\in L^1 \cap L^2$ then $$\|f\|_2 = \|\hat f\|_2$$ where $\hat f$ is the Fourier transform of $f$. On the ...
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### What's the idea of computing sums in Fourier series using said Fourier series (and Parseval's id)?

What's the idea of computing sums in Fourier series using said Fourier series (and Parseval's id)? I'm asked to find a value of a series using a Fourier series. But I have not material explaining how ...
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### Compute $\sum_{n=1}^\infty{\frac{1}{n^8}}$ using Parseval's Theorem

I need to show that $$\sum_{n=1}^\infty{\frac{1}{n^8}} = \frac{\pi^8}{9450}$$ I have already shown that $$\sum_{n=1}^\infty{\frac{1}{n^4}} = \frac{\pi^4}{90}$$ by computing the Fourier series for the ...
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### Sum of 1/n^4 using a half period cosine series

I am aware that I can solve the $$\sum_{n=1}^\infty\frac{1}{n^4},$$ using a a cosine series for $x^2$ on the half period $0<x<2$ however I am wondering if I can also solve this by using the ...
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