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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How to verify Parseval's theorem

I'm wondering how to verify the Parseval's theorem $$\int_{-\infty}^{\infty} f(x)g(x)dx = \int_{-\infty}^{\infty} F(k) G^{*}(k)dk$$ for 2 functions $f(x)$ and $g(x)$ I know how to find the Fourier ...
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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 ...
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How does monotonicity guarantee that the Fourier series is exactly convergent to the given function (Parseval's identity)?

In Piskunov's calculus, Bessel inequality is proved as follows: They start with a periodic function $f(x)$ having period $2\pi$ whose Fourier coefficients are $(a_r,b_r)$. Suppose $$T_n:=\{ s_n(x) = \...
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2 answers
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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 ...
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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 ...
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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 ...
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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{...
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$\sum_{k=1}^\infty\frac{1}{k^4} = \frac{\pi^4}{90}$ 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 $$\frac{1}{12}+\sum_{k\in \mathbb{N}}\frac{1}{\pi^2 k^2}$$...
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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 ...
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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\...
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Generalize Power spectrum density to higher orders

If the power spectrum density for a given time series is given by : $PSD(f) = \frac{2 \Delta}{N} \sum_{j=1}^{N} \delta B^{2}(t_{j}, f)$, where, where $δB(t_{j} , f)$ is the magnitude of the trace ...
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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(\...
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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 ...
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Example of a maximal orthonormal set in a non-Hilbert space

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 = ...
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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 ...
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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: $$\...
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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 ...
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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 ...
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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 $\...
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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{\...
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3 votes
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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}^{*}...
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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 $ \...
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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 ...
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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 ...
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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} \...
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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 ...
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2 votes
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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 ...
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1 answer
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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}...
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1 answer
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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[\...
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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)\:...
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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 $...
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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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Use Parseval's identity for series with $\sin((2k+1)x)$.

I'm trying to use Parseval's identity to evaluate the values of the series $$\sum_{k=0}^{\infty}\frac{1}{(2k+1)^6}$$ using a Fourier series that I have derived earlier as $f(x)=x(\pi-|x|) = \sum_{k=0}^...
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Factoring inequalities using Iverson identity - confused by double summations in Concrete Mathematics book

In chapter 2 section 4 (multiple sums) of Concrete Mathematics(Graham,Knuth,Patashnik) the authors use Iverson Identity to rearrange the variables' bounds. In particular, they start off with a ...
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Apply Parseval equality, to show that $f_h \in L^2[-\pi,\pi]$

Let $f$ a complex function, $f \in L^2[-\pi,\pi]$. Let $c_n=\frac{1}{2\pi} \int_{-\pi}^{\pi}f(x)e^{-inx}dx$ the Fourier coefficients, $n \in \mathbb{Z}$. Problem 1. Find fourier coefficients $\{c_n(h)\...
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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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Fourier series and Parseval-Bessel formula

I parametrized the boundary $\Gamma$ of the upper half unit disc by the following manner: $$\gamma(\theta)=\gamma(e^{i\theta})=e^{i\theta} \text{ if }\theta \in [0,\pi],$$ $$\gamma(t)=e^{i(t+3)\pi/2}...
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Fourier Coefficients and an infinite sum

I've been given the piecewise function (on period interval $]-\pi,\pi]$ $$f(x) = \begin{cases} \pi & x=\pi \\ x & 0\leq x < \pi \\ 0 & -\pi<x<0 \end{cases} $$ ...
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Arc length of a curve bounding from below a Fourier series

Assume $\gamma $ is a $C^1$ closed curve in the complex plane whose length is $2\pi$,and consider all its possible regular parametrization throught a parameter $t \in [0,2\pi]$ Let $\gamma(t)$ one of ...
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Computing $\sum_{k=1}^\infty\frac{1}{(2k-1)^4}$ with complex Fourier series and Parseval's equality

Let $f(x)=|x|$ I wish to compute corresponding complex Fourier coefficients $c_k$ given by $$ c_k=\frac{1}{2T}\int_{-T}^Tf(x)e^{-ikx} dx $$ for $T=\pi$. And then use the Parsevalle's equality, which ...
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1 vote
1 answer
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Orthonormal basis and prove $\langle\phi_\beta(x), \phi_\beta(y)\rangle'=\langle[x]_\beta, [y]_\beta\rangle' = \langle x,y\rangle$

Let $\{v_1, v_2, \ldots, v_n\}$ be an orthonormal basis for a finite-dimensional inner product space $V$ over some field $F$. For any $x, y$ in $V$, $\langle x, y \rangle = \sum\limits_{i = 1}^n ...
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Parseval’s Identity

In this proof of Parseval’s Identity, what is the justification for making the assumption $g(-t) = \overline{f(t)}$ and how does one obtain $g(t) = \overline{f(-t)}$ from $g(-t) = \overline{f(t)}$?
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Deriving Parseval relation from Parseval identity.

Consider a given orthonormal $M$ in a Hilbert space $H$. Then $M$ is total iff $$\sum_k |\left< x,e_k \right>|^2=\|x\|^2$$ holds for all $x\in H$ where $\left< x,e_k \right>$ are the ...
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2 answers
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Which of the following inequalities hold?

Let $f(z) = \large\sum_\limits{n=0}^{\infty}\normalsize a_n z^n\:$ be an entire function and let . Which of the following inequalities holds? $1.\sum _{n=0}^{\infty }\left|a_n\right|^2r^{2n}\le \...
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1 answer
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Use of Parseval’s theorem for deduction

How would you use Parseval’s theorem to show that $$\int_{0}^{\infty} \frac{1}{(k^2+1)^2}=\frac\pi4$$
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1 vote
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rigorous use of parseval identity for derivative to prove an inequality

Let $H$ be the Hilbert space generated by $$ \varepsilon =\left\{\sqrt{\frac{2}{T}} \, \cos\left( \frac{2\pi kt} T \right),\sqrt{\frac{2}{T}} \, \sin\left( \frac{2\pi kt} T \right) \right\} $$ using ...
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Parseval-Plancherel theorem for the Dirac delta

My question overlaps somewhat with: Is Plancherel's theorem true for tempered distribution? I am trying to better understand the answer provided there, and have some additional questions. Let $S$ ...
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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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2 votes
1 answer
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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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6 votes
2 answers
305 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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