# Questions tagged [parsevals-identity]

This tag is for questions regarding Parseval's Identity, an important result in the study of Fourier Series.

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### Relation between $\mathcal{L}_2$-norm of a signal and its amplitude in the frequency domain

Suppose for the signal $x(t)$, we know that $\|x\|_{\mathcal{L}_2}\le 1$, with this norm defined as $\|x\|_{\mathcal{L}_2}:= \sqrt{\int_0^{\infty}\|x(t)\|^2\mathrm{d}t}$, where $\|\cdot\|$ is the ...
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### Equality using Parseval's theorem?

Is it the following allowed using Parseval's theorem? $$\int_{\infty}^{\infty} g(k) f(x)^2 dx = \frac{1}{2\pi}\int_{\infty}^{\infty} g(k) F(k)^2 dk$$ with $F(k)$ the Fourier transform of $f(x)$.
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### Convergence of a sum similar to Parseval's Indentity

Suppose that $\{e_n\}$ is a othonormal basis for $L^2[0,T]$ and $\langle \cdot,\cdot \rangle$ be the standard $L^2[0,T]$ inner product. Denote $1_A(x)$ as the indicator function on the set $A$. Prove ...
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