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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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Absolute Convergence of Fourier Series Proof

I am looking at the following theorem Let $f$ and $g$ be two piecewise continuous, periodic functions with the same period $p$ and with Fourier series $$\begin{align} \mathcal{F}[f]|_{t} = \sum_{k=-\...
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Log Cepstral Distance proof using Parseval's theorem

The magnitude spectrum of $x(n)$ is given by $$ S(w) = \sum_{n = -\infty }^{+\infty} x(n) \cdot e^{-jwn} $$ The $log S(w)$ can be expressed as $$ log S(w) = \sum_{n = -\infty }^{+\infty} c_n \cdot e^{-...
Anantha Krishnan's user avatar
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Basel problem using $f(x)=x$ on $[0,2\pi)$

So I need to verify $$\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ Using Parsevals identity on $f(x)=x$ on $[0,2\pi)$. Does this mean if $a_n$ are the Fourier coefficients then $$\sum_{n=-\infty}...
homosapien's user avatar
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Using Parseval's to Evaluate an Integral

I have done some exercises on the Parseval's identity and I think it's quite straight forward. However, I came across this exercise and it made me confused. I'll explain: The function is $f(x) = \pi x ...
Zeeko's user avatar
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Extending Parseval's Equality

I wish to extend Parseval's Equality (or disprove that the following doesn't hold): Conjecture. Let $f,g\in L^1[0,1)$ such that $f\overline{g}\in L^1[0,1)$. Then $$ \int_0^1 f(x)\overline{g(x)}\,dx = \...
Doofenshmert's user avatar
4 votes
1 answer
163 views

Parseval-Plancherel type identity for probability generating function

Assume that $f,g \in L^2(\mathbb R)$ and define the Fourier transform of $f$ by $$\hat{f}(\xi) = \int_{\mathbb R} \mathrm{e}^{-i\,x\,\xi}\, f(x)\,\mathrm{d}\xi, \quad \xi \in \mathbb R.$$ The well-...
Fei Cao's user avatar
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Parseval's identity for Hilbert space-valued function.

Let $\mathcal H$ be a (separable) Hilbert space on $\mathbb C$. Suppose $f\in L^2([-\pi,\pi), \mathcal H)$ and $c_n(f)$ is n-th Fourier coefficient of $f$, i.e., $c_n(f)=\frac{1}{2\pi}\int_{-\pi}^\pi ...
daㅤ's user avatar
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2 votes
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Evaluation of Fourier series $\sum_{n=1,3,5...} \left[\frac{1}{n}\text{e}^{-\frac{n \pi x}{a}} \text{sin}(\frac{n \pi y}{a}) \right]$

I was studying electromagnetism and followed 'Introduction to Electromagnetism' by David Griffiths. During his derivation of the solution to Laplace's equation in ch. 3.3, he derives the equation $$V(...
Rasmus Andersen's user avatar
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Parseval's (Plancherel's) identity for Hilbert space-valued operator.

Let $\mathcal H$ be a (separable) Hilbert space on $\mathbb C$. Suppose $f\in L^2([-\pi,\pi), \mathcal H)$ and $c_n$ is n-th Fourier coefficient of $f$, i.e., $c_n=\frac{1}{2\pi}\int_{-\pi}^\pi e^{-in ...
daㅤ's user avatar
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Using Parseval's identity to evaluate a symmetric infinite sum

Let $f(x) = e^{i\alpha x}, x\in(-\pi,\pi)$ where $\alpha$ not an integer and an orthogonal basis $\{e^{ikx}\}$, show that $$\sum_{n\in\mathbb{Z}} \frac{1}{(n+\alpha)^2} = \frac{\pi^2}{\sin^2(\alpha\pi)...
860009898987's user avatar
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Maximal Overlap Wavelet Transform Energy Conservation Properties

When I perform the haar maximal overlap wavelet transform of a signal I get a series of coefficients and one approximation of the signal itself. ...
Emiliano Rosso's user avatar
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Calculate the sum of $\lim_{N\rightarrow \infty} \sum_{-N}^N 1/(n+e)$

Problem Calculate the sum of $S = \lim_{N\rightarrow \infty} \sum_{-N}^N 1/(n+e)$ My attempt We quickly realize that $\frac{1}{e+n} + \frac{1}{e-n} = \frac{2e}{e^2-n^2} = \frac{-2e}{n^2+(ie)^2}$. ...
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Finding a closed expression for the Riemann zeta function at even inputs

The problem I'm trying to derive a closed form expression for the values of the Riemann Zeta function for even natural numbers $k \geq 2$, meaning we want to find a closed form expression for $\zeta(k)...
Tanamas's user avatar
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Using Parseval's Identity to Calculate an Integral

The exercise is to use Parseval's identity to solve the following integral: \begin{equation} \int_{-\pi}^{\pi}|\sum_{n=1}^{\infty}\frac{1}{2^n}e^{inx}|^2 dx \end{equation} Now, I know that the ...
Zeeko's user avatar
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Show that $L^2$-norm of function and its Fourier transform coincide

Let $d\in\mathbb N$ and $D\in\mathcal B(\mathbb R^d)$ be bounded. Moreover, let $\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$ and $$\hat f(\omega):=\int e^{-{\rm i}2\pi\langle\omega,...
0xbadf00d's user avatar
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Prove that an orthonormal set $\{y_1,y_2,y_3,\dots\}$ is an orthonormal basis iff $\sum_{m=1}^\infty \vert(y_m,x_n)\vert^2=1\forall n\in \mathbb N$

Let $\{x_1,x_2,x_3,\dots\}$ be an orthonormal basis for the Hilbert Space $H$ over the field $\mathbb R$ with inner product $(\;\;,\;\;)$. Then, prove that an orthonormal set $\{y_1,y_2,y_3,\dots\}$ ...
Sayan Dutta's user avatar
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Proof of Plancherel's Theorem

This exercise suggests an alternative proof of Plancherel's Theorem for $L^2(\mathbb{T})$. Given $f \in L^2(\mathbb{T})$, define $f^*(\theta) := \overline{f(-\theta)}$. (a) Prove that for any ...
Sayan Dutta's user avatar
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Calculating $\sum_{k = 1}^{\infty} k^{-8}$ using Parseval´s identity

I calculated the Fourier series of $x^2$ getting $$x^2=\frac{\pi^2}{3}+4\sum_{k=1}^\infty \frac{(-1)^k}{k^2} \cos(kx)$$ Then, integrating this equation two times i got $$\frac{x^4-2\pi^2x^2}{12}=4\...
Lorentz's user avatar
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Calculate $\sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^2}$ using Parseval

The exercise asks me to calculate $\displaystyle{\sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^2}}$ using the Fourier series of \begin{equation} f(x) = \begin{cases} 0 & -\pi < x < 0 \\ ...
Merkel_Bot's user avatar
3 votes
2 answers
1k views

Use parseval's identity to prove $\int_0^\infty {\sin^4t \over t^2} dt = {\pi \over 4}$

I know how to calculate $$\int_0^\infty {\sin^4t \over t^4} dt$$ by taking the function $f(x)=1-|x|$ and it will be $\pi \over 3$. But here the denominator is not raised to power $4$ but $2$. How ...
Anshul Bishnoi's user avatar
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1 answer
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changing order of integration and sum

Suppose I have a function that is represented as follows $$ f(x) = \sum_{n=1}^{\infty} a_n(x) $$ for $- \pi \leq x \leq \pi$. Then is it always the case that $$ \int_{-\pi}^{\pi}f^2(x) dx = \sum_{n=1}...
Johnny T.'s user avatar
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3 votes
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Parseval's Identity Application

In this video, it is stated that Parseval's Identity "is how we go from discrete to continuous." However, I have not been able to find any material that expands on this use of Parseval's ...
user10478's user avatar
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206 views

Show that $\int_{-\infty}^{\infty} \frac{\sin(2\pi(x-a))}{\pi(x-a)}\frac{\sin(2\pi(x-b))}{\pi(x-b)}dx = \frac{\sin2\pi(a-b)}{\pi(a-b)}$

I want to show that for any $a,b \in \mathbb{R}$ we get $\int_{-\infty}^{\infty} \frac{\sin(2\pi(x-a))}{\pi(x-a)}\frac{\sin(2\pi(x-b))}{\pi(x-b)}dx = \frac{\sin2\pi(a-b)}{\pi(a-b)}$. A hint for this ...
TOMILO87's user avatar
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1 answer
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Using Parseval's Identity

Using parsevals Identity we have obtained that $$t = \sum_{n=-\infty}^\infty \frac{i(-1)^ne^{-int}}{n} $$, and $c_0=0$ , prove that $\frac{\pi^2}{6}= \sum\frac{1}{n^2}$. I am really struglling this ...
ben huni's user avatar
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1 answer
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Limit of integral with Parseval

I am trying to evaluate the limit $$ \lim_{n \to \infty} \int_{-\infty}^{\infty} \frac{\sin(nt)}{\pi t}f(t) \,dt, $$ where $$ f(t)=e^{-t^2+2t}. $$ Since I am working with Fourier transforms I thought ...
matte_studenten's user avatar
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655 views

Parseval's identity on Fourier series of $f(x)=e^x$

Let $\{\varphi_k\}_{k=1}^\infty$ be an orthogonal system, and $\{\alpha_k(f)\}$ the Fourier coefficients for a function $f\in L^2([a,b])$. Then the Parseval's identity is given by the formula \begin{...
GreekCorpse's user avatar
1 vote
1 answer
659 views

$\sum_{k=1}^\infty\frac{1}{k^4} = \frac{\pi^4}{90}$ using Parseval's Theorem and Fourier series

Prove $$\sum_{k=1}^\infty\frac{1}{k^4} = \frac{\pi^4}{90}$$ using Parseval's Theorem and Fourier Series of $$f(x)=(x-\frac{1}{2})^2$$ which is $$\frac{1}{12}+\sum_{k\in \mathbb{N}}\frac{1}{\pi^2 k^2}$$...
Gunners 's user avatar
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Is the identity $(\chi_V,\chi_V)=\sum_{i=1}^k a_i^2$ appearing in character theory related to Parseval’s identity from Fourier analysis?

I’m taking an introductory course on finite group representations/character theory and just read about this: Given a representation $\rho:G\rightarrow V$ of a group $G$ with associated isotypic ...
dahemar's user avatar
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How to use the Parseval's theorem to show this relationship?

Problem My goal is to show the following relationship holds: $$ \mathcal{S} \triangleq \frac{\int\int_{-\infty}^{\infty}P(x,y)e^{j\phi(x,y)}dxdy} {\int\int_{-\infty}^{\infty}P(x,y) dx dy} =\frac{\int\...
Ogiad's user avatar
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Parseval identity

I am reading LECTURE NOTES 2 FOR 247A,TERENCE TAO. At the beginning of the paragraph the author defines the fourier trasform $\mathcal{F}f$ of a function $f$ and the hypothesis is that $f \in L^1(\...
Lau's user avatar
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6 votes
1 answer
745 views

Sum of $\dfrac{1}{n^2+1}$ using Parseval's theorem

I know this question has been widely answered here, but without using Fourier analysis. Also there is a video referring to this trick but I want to use a different Fourier series. First of Parseval's ...
Leon's user avatar
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0 votes
1 answer
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Example of a maximal orthonormal set in a non-Hilbert space [duplicate]

We know that the following statements are equivalent in a Hilbert space for an orthonormal set $\left\lbrace e_{\alpha} | \alpha \in \Delta \right\rbrace$: For each $x \in H$, we have $$\| x \|^2 = ...
Aniruddha Deshmukh's user avatar
1 vote
1 answer
368 views

Deduce from Parseval's equality $\frac{\pi^2}{\sin^2(\pi x)}=\sum_{n=-\infty}^\infty\frac{1}{(x-n)^2}\,,\forall x\in\mathbb{R}\setminus\mathbb{Z}$

Exercise 4.6 in Gasquet and Witomski (1999) [1]: Find the Fourier expression of the function $f$ with period 2 defined on $[-1,1)$ for $z\in\mathbb{C}\setminus\mathbb{Z}$ by $$ f(t) = \exp\left\{i\pi ...
javlacalle's user avatar
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Clarification or guidance on exercise involving integral transform

I have an exercise that I'm running circles around, and I'd like to state the problem, then discuss what I've attempted, and ask for some guidance. The problem is to prove the following equality: $$\...
charlesFL's user avatar
1 vote
1 answer
905 views

Prove the series formula for $\frac{\pi^4}{96}$ using Parseval's Identity

I have seen this post Fourier series expansion of $\frac{\pi^4}{96}$ and $\frac{\pi^4}{90}$ but it seems to skip some steps that I don't understand. I also looked at some others, but I haven't found ...
Nolan P's user avatar
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Parseval Identity and Fourier Coefficients

I have to calculate Fourier coefficients and write Parseval Identity of : $ f(x)$ defined as: $ f(x)=cos^2(x) $ if $ -\pi/2\le x\le\pi/2$ and $ f(x)=0$ otherwhise in $L^2(-\pi, \pi)$, in the base of ...
Shanks Tyler Red's user avatar
1 vote
1 answer
168 views

Hilbert space representation in $H_0^1$

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain and let $\partial\Omega$ be of class $C^2$. Let $v_k$ be the $k$-th eigenfunction of $-\Delta v=\lambda v$ and boundary condition $v=0$ on $\...
user99432's user avatar
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2 votes
0 answers
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How to compute fourier transform of the function knowing its spectral density function?

I have the following function (where W denotes displacement of cylindrical shell under some force): $S_w(x_1, x_2) = W(x_1, x_2)W(x_1, x_2)^* = \sum_{m_1, m_2 = 1}^{\infty}C(m_1, m_2) \cdot\sin^2{\...
Igor Kotua's user avatar
3 votes
1 answer
94 views

Compute $\int_{-\infty}^{\infty} t \tan^{-1}(t) \exp (-t^2)\,dt$

I would like to obtain $$ \int_{-\infty}^{\infty} t \tan^{-1}(t) \exp (-t^2)\,dt $$ My idea is to use Fourier transform and go with generalized Parseval. I choose $x_{1}(t)=\tan^{-1}(t)$ and $x_{2}^{*}...
why_me's user avatar
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1 vote
3 answers
129 views

Evaluate integral $ \int_{\mathbb{R} } \frac{\sin^4(\frac{t}{2}) }{t^2}$

Let $f(x)= (1-2|x|)\chi_{[-\frac{1}{2}, \frac{1}{2}]} (x) $. Its Fourier transform is given by $ \hat{f} (x) = \frac{8\sin^2(\frac{t}{4})}{t^2} $. Based on this, I need to evaluate the integral $ \...
user121's user avatar
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0 votes
0 answers
102 views

Calculating an integral using fourier transform.

I am trying to show that $\int_0^{\infty}{\frac{\sin(x)}{x}} = \frac{\pi}{2}$ using the fourier transform of $f(x) = \sin(ax)/\pi x$. I found using symmetry formula and the transform of a rectangular ...
Governor's user avatar
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1 answer
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Application of Parseval's identity to a vectorial series

Let $H$ be a Hilbert space and let $\{e_n, n\in \mathbb N\}$ be an orthonormal sequence in $H$. Determine whether these series converge in $H$: $\sum \frac{e_n}{n}$ $\sum\frac{e_n}{\sqrt{n}}$ The ...
Furdzik Zbignew's user avatar
1 vote
0 answers
76 views

Parseval's relation

I've been reading Gut's book for probability theory class. I got stuck on the problem 12 from Chapter 10. (p.198) I don't know how can I proof via Parseval's relation that: $$\int_{-\infty}^{\infty} \...
christk's user avatar
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0 answers
120 views

What is the correct formula for Parseval's identity?

Looking for a quick validation here: We have a periodic function $f(t)$ with a period of $T$ that satisfies Dirichlet's conditions. Which of the following 2 formulas is correct ? My book (not ...
NickDelta's user avatar
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2 votes
1 answer
165 views

Brownian Motion construction - How is Perseval's identity applied? Why does the below fact imply the existence of limit?

In Schilling, Partzsch, referring to Levy-Ciesielski construction of Brownian Motion, I read that: [...] idea is to write the paths $[0,1]\ni t \mapsto B_t(\omega)$ for almost every $\omega$ as a ...
Strictly_increasing's user avatar
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1 answer
99 views

Why this is Kronecker-delta?

I was trying to prove this theorem: If $f(x) = \int_{-\infty}^{\infty} c_ne^{in\pi x/L}$, then show that $\langle|f(x)|^2\rangle = \sum |c_n|$ I checked the solution: $\langle|f(x)|^2\rangle = \frac{1}...
Z. Huang's user avatar
3 votes
1 answer
89 views

How to show $ \sum_{n=1}^{\infty}\left(\frac{\sin (n b)}{n}\right)^{2}=\frac{b \pi-b^{2}}{2} $ - Parseval's identity?

I have trouble solving a). How do I approach this problem? Let $-\pi\leq a<b\leq\pi$. Consider the function $$f(x)=\left\{\begin{array}{ll}1, & x \in] a, b[ \\ 0, & x \in]-\pi, \pi[\...
mhj's user avatar
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0 answers
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Help understanding this passages about Fourier transforms

Let $$\Psi_{J,K,\theta}(x,y)=\frac{1}{\sqrt{J}}\Psi\left(\frac{\cos(\theta)x+\sin(\theta)y-K}{J}\right)$$ Let $$\mathcal{T}[f](J,K,\theta)=\int_{\mathbb{R}^2}\overline{\Psi_{J,K,\theta}(x,y)}f(x,y)\:...
riemannfanboy's user avatar
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1 answer
481 views

Using Parseval's identity to evaluate a definite integral.

Using the Parseval’s identity prove that $$\int_{-\pi}^\pi\cos^{4}(x)dx= \frac{3\pi}{4}$$ As far as I know to do this problem we need to find the fourier coefficients of $\cos^{2}(x)$. I am getting $...
Clarence Callahan's user avatar
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0 answers
75 views

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 $f(x)$.
Fre's user avatar
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