Questions tagged [parsevals-identity]

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Generalizations to Parseval theorem

I was wondering if there are any generalizations to the Parseval's theorem in the case of overcomplete representations. I'm a bit of a novice in advanced linear algebra, so please bear with me. Let $\...
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1answer
31 views

Parseval's identity with complex Fourier

Define the Fourier coefficients of a function to be $\hspace{2cm}\mathscr{(F}f)(n)=\frac{1}{2\pi}\int_{0}^{2\pi}f(x)e^{-inx}dx$. (This is $c_{n}$ in the book, but we indicate explicit dependence on ...
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1answer
26 views

Using Parsevals formula to calculate a sum

I have: $$g(x)=x(1-|x|), \space\space\space\space\space\space -1\leq x \lt 1$$ From which I got the Fourier series: $$Sg(t)=\frac{-8}{\pi^3}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^3}\sin((2n+1)x)$$ The ...
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1answer
116 views

show this nice trigonometric identiy

I was working on a book, which was asking me to prove that some product is equal to nn. I had reduced the problem to proving a trigonometric identity, but I couldn't prove it although I spent much ...
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37 views

If the integral of a function is known, can anything be said about products of integrals of the function and its Fourier transform?

Suppose $f \in \mathcal S(\mathbb R)$ is a Schwartz function. Then the Fourier transform is invertible, with nice properties like $$\int f(x) \overline{\hat g(x)} \mathrm{d} x = \int \hat f(x) \...
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29 views

Parseval Theorem for a finite-valued function to the p-th power

According to Parseval Theorem (or Plancherel theorem), we have the following property. If $f(x)$ and $g(x)$ are two $L^2$ functions, and $P$ denotes the Plancherel transform, $$ \int_{-\infty}^{\...
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27 views

Does generalized Parseval identity series: $\langle g,h\rangle =\sum_{1}^{\infty}\langle g,e_i\rangle\langle e_i,h\rangle$ absolutely converge

The generalized Parseval identity states that given $\{e_i\}_1^\infty$ a complete orthonormal system in Hilbert space $H$ then for all $g,h\in H : \langle g,h\rangle =\sum_{i=1}^{i=\infty} \langle g,...
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2answers
49 views

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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2answers
35 views

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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52 views

On conditions for Plancherel Theorem

If we have real functions such that $f \in L^1$ and $g \in L^2$, do we still have always the following equality, providing left hand-side is well defined (Plancherel theorem ?) : $$\int_{-\infty}^{\...
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299 views

How to prove this identity nicely?

Show that$$\sum_{1\le i<j\le n}\left((x_j-x_i)-(x_j-x_i)^2\right)=\left(\sum_{i=1}^n{x_i}\right)^2-n\sum_{i=1}^n{x_i^2}-\sum_{i=1}^n{(n-2i+1)x_i}\\=-n\sum_{i=1}^n\left(x_i-\frac1n\sum_{j=1}^n{x_j}+\...
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14 views

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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1answer
25 views

Parseval identity for Laurent series

Let $0\leq r_1<r_2$, and $z_0\in \mathbb{C}$, and consider the region $A=\{z\in \mathbb{C}|r_1<|z-z_0|<r_2\}$. Let $f$ analytic in the region $A$. Then we can write, $$f(z)=\sum_{n=0}^{\infty}...
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43 views

Sturm - Liouville Parseval proof

Consider the expansion $𝑓(𝑥)=\sum_{n=0}^{\infty}A_𝑛\phi_𝑛(𝑥)$ where $\phi_𝑛(𝑥)$ is the set of eigenfunctions of the Sturm-Liouville problem, and $𝑓(𝑥)$ is any piecewise differentiable ...
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236 views

Statement of Parseval's theorem for Fourier Transform

the following is the statement of Parseval's theorem from Wikipedia, Suppose that $A(x)$ and $B(x)$ are two square integrable (with respect to the Lebesgue measure), complex-valued functions on $\...
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178 views

Prove $\sum_{n=1}^\infty\frac{1}{(2n-1)^4}=\frac{\pi^4}{96}$ [closed]

Prove using Parseval identity applied to the functions: $x\,,|x|, x^2$ the convergence of the sum: $$\sum_{n=1}^\infty\frac{1}{(2n-1)^4}=\frac{\pi^4}{96}\tag1$$ My attempt: The identity of Parseval ...
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1answer
62 views

Extra factor of 2 when evaluating an infinite sum using fourier series and parseval's theorem.

I'm asked to find the fourier series of the $2 \pi $ periodic function f(x) which is $sin(x)$ between $0$ and $\pi$ and $0$ between $\pi$ and $2\pi$ I use the complex form to proceed and get $$\frac{...
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82 views

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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1answer
48 views

$L^2$ convergence of Fourier series

Let $R_n(x)=f(x)-S_n(x)$ where $S_n(x)$ is the partial sum of Fourier series of $f(x)$ function. $\lim_{n \to \infty} <R_n(x),R_n(x)>$ $=0$  $\iff$  $\sum_{n=1}^{\infty}c_i^2 = \int_{-\pi}^{\...
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28 views

integral and Fourier transform

I want to show that $\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{d}}f(x_{1},x_{2})f(y_{1},y_{2})\vert y_{2} -x_{2}\vert^{d-k} dx_{1}dx_{2}dy_{1}dy_{2}$ is finite knowing only the Fourier transform of f. $...
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1answer
127 views

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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1answer
147 views

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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1answer
388 views

Problem using Parseval's theorem for solving an integral

I need to use Parseval's theorem to calculate the following integral: $$\int_{-\infty}^{\infty}\left |\frac{1-e^{-iwt}}{iw} \right |^{2}dt$$ I thought to find the transform of $$f(t) = \frac{1-e^{-...
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1answer
62 views

Parseval equality with $\int |f|^{1}$

The famous Parseval equality states that \[ \int|f|^{2} = \int |F|^2, \] where $f$ and $F$ are related to one another by \[ \int f e^{-2 \pi i r y } dr = F(y). \] My question is, whether there is any ...
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33 views

Proving some identities using Fourier series

Find the fourier series of the following function: $$f:[-\pi,\pi]\to \mathbb{R} , f(x) = \pi x - x^3$$ Easy task. $a_n = 0$ since f is odd and $b_n = \dfrac{2\pi}{n}(-1)^{n+1} + \dfrac{2{\pi}^2}{n}(-...
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1answer
83 views

Parseval to find length [closed]

Use Parsevals identity to find the length $$|f| = \sqrt{\frac{1}{\pi} \int_{-\pi}^{\pi}[f(x)]^{2}dx}$$ for $f(x) = 2 \cos(14x) + 4 \cos(11x) + 2 \sin(27x) - \cos(19x) + 5 \cos(140x)$. So I'...
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1answer
420 views

Calculate Fourier series sum using Parseval's theorem

I've used Parseval's theorem before, to calculate $\sum_{n=1}^\infty \frac{1}{n^4}$. Now I have to calculate $ \sum_{n=1}^\infty \frac{1}{(2n-1)^4}$ and I'm lost. I have to start with $f(x)=x, x\in[0,\...
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1answer
52 views

Help with a step in the Parseval Theorem

I (think I) understand everything up until the step that integrates $(f(x))^2$ at $x \in [-\pi, \pi]$. I understand why the zero occurred, my understanding is that $\cos$ and $\sin$ cancel out after ...
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4answers
144 views

Fourier series analysis

Expand the following periodic signal in Fourier series: $$s(t)=2\sin (1000\pi t)+0.5\sin (500\pi t)+\cos (250\pi t), -\infty < t <+\infty.$$ Determine the basic period of that signal, mean ...
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123 views

Parseval's theorem and $L_2$ space

I need to show $\sum|\hat{f}(n)|< \infty$ if $f$ is continuously differentiable on the circle $\mathbb{T}$. Using the fact that $\hat{f}(n)= \frac{\hat{f}'(n)}{in}$, Cauchy-Schartz and Parseval's ...
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140 views

Integration using Parseval's theorem

Would it be possible, to have some guidance to compute the following, using Parseval's theorem for any integer N≥1. Thanks.
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73 views

Parseval's Identity Proof Monotone/Dominated Convergence Theorem

I proved Parseval's identity as follows: If $f\in L^2\left(\mathbb{R}/\mathbb{Z}\right)$, then \begin{equation*} \int\left|f\left(t\right)\right|^2\,dt=\sum_{n\in\mathbb{Z}}\left|c\left(n\right)\...
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1answer
271 views

Parseval's theorem rewritten in Fourier series

Given a function $f \in L^2(-\pi,\pi)$ Parseval's theorem states $$ \frac{1}{2\pi} \int_{-\pi}^{\pi}|f(x)|^2 dx=\sum_{n=-\infty}^{\infty}|c_n|^2$$ Is the following also true? $$ \frac{1}{2\pi} \int_{-...
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1answer
334 views

Parseval identity in FFT

I was recreating in Matlab a certain function using FFT. In particular I was interesting in knowing how modes are sufficient to approximate quite well my function. To do that, I calculate the norm ...
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2answers
341 views

using Parseval's identity to estimate the value of $\Sigma_{n=1}^\infty \frac{1}{(2n-1)^2(2n+1)^2}$

There is a problem with two parts; The first part is asking to find Fourier series for $f(x)=|\sin(x)|$ on $[-\pi,\pi]$. And the second part wants to estimate the following using Parseval's identity: $...
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2answers
385 views

Use Parseval's equation and the table of Fourier series to evaluate $\sum\frac{1}{(1+n^2)^2}$

Use Parseval's equation and the table of Fourier series to evaluate $$\sum\frac{1}{(1+n^2)^2}$$ So I have used this method before to show $$\sum\frac{1}{n^4} = \frac{\pi^4}{32}$$ however, for this ...
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1answer
112 views

Evaluating $\int_{\mathbb{R}}\left( \frac{\sin x}{x}\right )^3$ with Parseval's identity

If this were an even power it would be clear enough how to approach this, although I wouldn't be crazy about computing the Fourier transform. For reference: I was able to solve $$ \int_{\mathbb{R}}\...
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2answers
375 views

Confusion on Parseval's Theorem

Parseval's Theorem says that: $$\int_{-\infty}^{\infty}g(t)f(t)^\ast dt = \frac{1}{2\pi} \int_{-\infty}^{\infty}G(\omega)F(\omega)^\ast d\omega$$ Although I know how to prove it, it's difficult to ...
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0answers
112 views

Sum Identity without Parseval's Theorem

Show that $\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$ The following Sum Identity was derived by applying Parseval's Theorem to the integral in the question linked above. $\...
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2answers
466 views

Parseval's theorem/Identity - Definition seems wrong

My book shows with some steps that $$\int_{-L}^L {f(x)}^2dx=\int_{-L}^L\left\{\frac{1}{2}a_0+\sum_{n=1}^\infty\left[a_n\cos{\left(\frac{n\pi x}{L}\right)}+b_n\sin{\left(\frac{n\pi x}{L}\right)}\right]\...