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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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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if f(x) = cosx, g(x) = e^-|x|, how can I use them to proof parseval theorem.

the theorem is integral x belongs to R f(x)g^hat(x)dx = (1/2pi)*integral w belongs to R F(w)G^hat(w)dw. From left hand side, the result is 1, but I don't know how to calculate the right hand side.
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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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53 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 ...
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1answer
40 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}...
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40 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[\...
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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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67 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 $...
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38 views

sum of the square of binomial coefficient [duplicate]

Using Vandermonde's identity, I can have $$ \sum_{i=0}^{n} \binom{n}{i}^2=\binom{2n}{n} $$ Is there a closed expression for $$ \sum_{i=0}^{n} \binom{n}{i}^2 x^{i}. $$ If we define a function $$ f(x)=\...
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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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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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24 views

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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27 views

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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Parseval's identity for 2D function

Can anyone please give an expression to the Parseval's identity for 2D function as shown here: Real-valued 2D Fourier series? Thanks a lot
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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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1answer
43 views

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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40 views

Parseval's Formula Proof

Let $f$ be continuous on $[0,2\pi]$ with period $2\pi$. Let $f(x)\sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}(a_{n}cosnx+b_{n}sinnx)$ be the Fourier series generated by $f$, and let $s_{n}(x)$ be the nth ...
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52 views

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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1answer
74 views

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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200 views

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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54 views

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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57 views

Fourier Transform and Parseval's theorem

Could someone please demonstrate how to do the following: Consider a function F(t) which has a value of zero for negative t and, for t>0, a value $$ e^{\frac{-t}{2 \tau}}$$ Find its Fourier ...
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2answers
67 views

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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1answer
68 views

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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1answer
51 views

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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127 views

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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1answer
215 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
69 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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289 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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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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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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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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147 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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109 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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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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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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1answer
56 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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1answer
673 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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871 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
102 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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2answers
96 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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1answer
172 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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1answer
174 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
195 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
572 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
75 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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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}(-...