Questions tagged [parsevals-identity]
This tag is for questions regarding Parseval's Identity, an important result in the study of Fourier Series.
65
questions
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vote
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30 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:
$$\int_{-\infty}^{\infty} \...
0
votes
0answers
18 views
if f(x) = cosx, g(x) = e^-|x|, how can I use them to proof parseval theorem.
the theorem is integral x belongs to R f(x)g^hat(x)dx = (1/2pi)*integral w belongs to R F(w)G^hat(w)dw.
From left hand side, the result is 1, but I don't know how to calculate the right hand side.
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0answers
32 views
What is the correct formula for Parseval's identity?
Looking for a quick validation here:
We have a periodic function $f(t)$ with a period of $T$ that satisfies Dirichlet's conditions. Which of the following 2 formulas is correct ?
My book (not ...
2
votes
1answer
53 views
Brownian Motion construction - How is Perseval's identity applied? Why does the below fact imply the existence of limit?
In Schilling, Partzsch, referring to Levy-Ciesielski construction of Brownian Motion, I read that:
[...] idea is to write the paths $[0,1]\ni t \mapsto B_t(\omega)$ for almost every $\omega$ as a ...
0
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1answer
40 views
Why this is Kronecker-delta?
I was trying to prove this theorem: If $f(x) = \int_{-\infty}^{\infty} c_ne^{in\pi x/L}$, then show that $\langle|f(x)|^2\rangle = \sum |c_n|$
I checked the solution:
$\langle|f(x)|^2\rangle = \frac{1}...
3
votes
1answer
40 views
How to show $ \sum_{n=1}^{\infty}\left(\frac{\sin (n b)}{n}\right)^{2}=\frac{b \pi-b^{2}}{2} $ - Parseval's identity?
I have trouble solving a). How do I approach this problem?
Let $-\pi\leq a<b\leq\pi$. Consider the function
$$f(x)=\left\{\begin{array}{ll}1, & x \in] a, b[ \\ 0, & x \in]-\pi, \pi[\...
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0answers
15 views
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 $$\mathcal{T}[f](J,K,\theta)=\int_{\mathbb{R}^2}\overline{\Psi_{J,K,\theta}(x,y)}f(x,y)\:...
0
votes
1answer
67 views
Using Parseval's identity to evaluate a definite integral.
Using the Parseval’s identity prove that $$\int_{-\pi}^\pi\cos^{4}(x)dx= \frac{3\pi}{4}$$
As far as I know to do this problem we need to find the fourier coefficients of $\cos^{2}(x)$. I am getting $...
0
votes
0answers
38 views
sum of the square of binomial coefficient [duplicate]
Using Vandermonde's identity, I can have
$$
\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)=\...
0
votes
0answers
14 views
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 ...
0
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0answers
24 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 $f(x)$.
0
votes
1answer
61 views
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}^...
0
votes
1answer
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 ...
2
votes
0answers
45 views
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)\...
0
votes
2answers
40 views
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 ...
1
vote
0answers
27 views
Fourier series and Parseval-Bessel formula
I parametrized the boundary $\Gamma$ of the upper half unit disc by the following manner:
$$\gamma(\theta)=\gamma(e^{i\theta})=e^{i\theta} \text{ if }\theta \in [0,\pi],$$
$$\gamma(t)=e^{i(t+3)\pi/2}...
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0answers
19 views
Parseval's identity for 2D function
Can anyone please give an expression to the Parseval's identity for 2D function as shown here:
Real-valued 2D Fourier series?
Thanks a lot
2
votes
0answers
41 views
Fourier Coefficients and an infinite sum
I've been given the piecewise function (on period interval $]-\pi,\pi]$
$$f(x) = \begin{cases}
\pi & x=\pi \\
x & 0\leq x < \pi \\
0 & -\pi<x<0
\end{cases}
$$
...
0
votes
1answer
43 views
Arc length of a curve bounding from below a Fourier series
Assume $\gamma $ is a $C^1$ closed curve in the complex plane whose length is $2\pi$,and consider all its possible regular parametrization throught a parameter $t \in [0,2\pi]$
Let $\gamma(t)$ one of ...
0
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0answers
40 views
Parseval's Formula Proof
Let $f$ be continuous on $[0,2\pi]$ with period $2\pi$. Let $f(x)\sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}(a_{n}cosnx+b_{n}sinnx)$ be the Fourier series generated by $f$, and let $s_{n}(x)$ be the nth ...
1
vote
1answer
52 views
Computing $\sum_{k=1}^\infty\frac{1}{(2k-1)^4}$ with complex Fourier series and Parseval's equality
Let $f(x)=|x|$ I wish to compute corresponding complex Fourier coefficients $c_k$ given by
$$
c_k=\frac{1}{2T}\int_{-T}^Tf(x)e^{-ikx} dx
$$
for $T=\pi$. And then use the Parsevalle's equality, which ...
1
vote
1answer
74 views
Orthonormal basis and prove $\langle\phi_\beta(x), \phi_\beta(y)\rangle'=\langle[x]_\beta, [y]_\beta\rangle' = \langle x,y\rangle$
Let $\{v_1, v_2, \ldots, v_n\}$ be an orthonormal basis for a
finite-dimensional inner product space $V$ over some field $F$. For
any $x, y$ in $V$, $\langle x, y \rangle = \sum\limits_{i = 1}^n
...
0
votes
0answers
200 views
Parseval’s Identity
In this proof of Parseval’s Identity, what is the justification for making the assumption $g(-t) = \overline{f(t)}$ and how does one obtain $g(t) = \overline{f(-t)}$ from $g(-t) = \overline{f(t)}$?
0
votes
0answers
54 views
Deriving Parseval relation from Parseval identity.
Consider a given orthonormal $M$ in a Hilbert space $H$. Then $M$ is total iff
$$\sum_k |\left< x,e_k \right>|^2=\|x\|^2$$
holds for all $x\in H$ where $\left< x,e_k \right>$ are the ...
0
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0answers
57 views
Fourier Transform and Parseval's theorem
Could someone please demonstrate how to do the following:
Consider a function F(t) which has a value of zero for negative t and, for t>0, a value
$$ e^{\frac{-t}{2 \tau}}$$
Find its Fourier ...
0
votes
2answers
67 views
Which of the following inequalities hold?
Let $f(z) = \large\sum_\limits{n=0}^{\infty}\normalsize a_n z^n\:$ be an entire function and let . Which of the following inequalities holds?
$1.\sum _{n=0}^{\infty }\left|a_n\right|^2r^{2n}\le \...
0
votes
1answer
68 views
Use of Parseval’s theorem for deduction
How would you use Parseval’s theorem to show that
$$\int_{0}^{\infty} \frac{1}{(k^2+1)^2}=\frac\pi4$$
1
vote
1answer
51 views
rigorous use of parseval identity for derivative to prove an inequality
Let $H$ be the Hilbert space generated by
$$
\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\}
$$
using ...
1
vote
0answers
127 views
Parseval-Plancherel theorem for the Dirac delta
My question overlaps somewhat with: Is Plancherel's theorem true for tempered distribution?
I am trying to better understand the answer provided there, and have some additional questions.
Let $S$ ...
0
votes
1answer
215 views
Parseval's identity with complex Fourier
Define the Fourier coefficients of a function to be
$\hspace{2cm}\mathscr{(F}f)(n)=\frac{1}{2\pi}\int_{0}^{2\pi}f(x)e^{-inx}dx$.
(This is $c_{n}$ in the book, but we indicate explicit dependence on ...
2
votes
1answer
69 views
Using Parsevals formula to calculate a sum
I have: $$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)$$
The ...
8
votes
2answers
289 views
show this nice trigonometric identiy
I was working on a book, which was asking me to prove that some product is equal to nn. I had reduced the problem to proving a trigonometric identity, but I couldn't prove it although I spent much ...
2
votes
0answers
47 views
If the integral of a function is known, can anything be said about products of integrals of the function and its Fourier transform?
Suppose $f \in \mathcal S(\mathbb R)$ is a Schwartz function. Then the Fourier transform is invertible, with nice properties like
$$\int f(x) \overline{\hat g(x)} \mathrm{d} x = \int \hat f(x) \...
1
vote
0answers
43 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,
$$
\int_{-\infty}^{\...
0
votes
2answers
79 views
Does generalized Parseval identity series: $\langle g,h\rangle =\sum_{1}^{\infty}\langle g,e_i\rangle\langle e_i,h\rangle$ absolutely converge
The generalized Parseval identity states that given $\{e_i\}_1^\infty$ a complete orthonormal system in Hilbert space $H$ then for all $g,h\in H : \langle g,h\rangle =\sum_{i=1}^{i=\infty} \langle g,...
1
vote
2answers
147 views
Question on Parseval's Theorem and Plancherel’s formula
I've come across Parseval's theorem and Plancherel’s formula several times on this forum. Each time they're referenced they're mentioned in regards to inner products in general. However, every proof I ...
0
votes
2answers
109 views
Proving parseval identity for trigonometric polynomials
Show that
$$||P||_2^2=\sum_{k=-N}^N \langle P,e_k\rangle^2,$$
where $e_k$ are the Fourier basic functions, and $P$ is a trigonometric polynomial of degree $N.$
I am not sure how to link trigonometric ...
1
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0answers
78 views
On conditions for Plancherel Theorem
If we have real functions such that $f \in L^1$ and $g \in L^2$, do we still have always the following equality, providing left hand-side is well defined (Plancherel theorem ?) :
$$\int_{-\infty}^{\...
4
votes
5answers
325 views
How to prove this identity nicely?
Show that$$\sum_{1\le i<j\le n}\left((x_j-x_i)-(x_j-x_i)^2\right)=\left(\sum_{i=1}^n{x_i}\right)^2-n\sum_{i=1}^n{x_i^2}-\sum_{i=1}^n{(n-2i+1)x_i}\\=-n\sum_{i=1}^n\left(x_i-\frac1n\sum_{j=1}^n{x_j}+\...
0
votes
1answer
56 views
Parseval identity for Laurent series
Let $0\leq r_1<r_2$, and $z_0\in \mathbb{C}$, and consider the region $A=\{z\in \mathbb{C}|r_1<|z-z_0|<r_2\}$. Let $f$ analytic in the region $A$. Then we can write, $$f(z)=\sum_{n=0}^{\infty}...
0
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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 $\...
-1
votes
1answer
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 ...
1
vote
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 $$\frac{...
2
votes
2answers
96 views
Hypotheses on Plancherel's theorem
Plancherel's theorem is stated as (e.g. in Rudin's Real and Complex Analysis)
If $f\in L^1 \cap L^2$ then
$$ \|f\|_2 = \|\hat f\|_2 $$
where $\hat f$ is the Fourier transform of $f$. On the ...
3
votes
1answer
172 views
$L^2$ convergence of Fourier series
Let $R_n(x)=f(x)-S_n(x)$ where $S_n(x)$ is the partial sum of Fourier series of $f(x)$ function.
$\lim_{n \to \infty} <R_n(x),R_n(x)>$ $=0$ $\iff$ $\sum_{n=1}^{\infty}c_i^2 = \int_{-\pi}^{\...
5
votes
1answer
174 views
Compute $\sum_{n=1}^\infty{\frac{1}{n^8}}$ using Parseval's Theorem
I need to show that $$\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}$$ by computing the Fourier series for the ...
0
votes
1answer
195 views
Sum of 1/n^4 using a half period cosine series
I am aware that I can solve the $$\sum_{n=1}^\infty\frac{1}{n^4},$$ using a a cosine series for $x^2$ on the half period $0<x<2$ however I am wondering if I can also solve this by using the ...
2
votes
1answer
572 views
Problem using Parseval's theorem for solving an integral
I need to use Parseval's theorem to calculate the following integral:
$$\int_{-\infty}^{\infty}\left |\frac{1-e^{-iwt}}{iw} \right |^{2}dt$$
I thought to find the transform of $$f(t) = \frac{1-e^{-...
0
votes
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 ...
1
vote
0answers
38 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}(-...
