This tag is for questions regarding Parseval's Identity, an important result in the study of Fourier Series.
Questions tagged [parsevals-identity]
106 questions
7
votes
1 answer
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 Theorem states:…
Leon
- 1,137
7
votes
1 answer
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 function $f(x)=x^2$ on the interval $(0,l)$ and…
mathqueen459
- 1,233
6
votes
2 answers
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 time on it. Then, I checked the solution, and it…
math110
- 95,328
- 17
- 156
- 531
4
votes
1 answer
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 $f(x) = \pi x - x^2$ on the interval [0, ]. I shall find the…
Zeeko
- 117
4
votes
1 answer
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-known Parseval-Plancherel identity links the $2$…
Fei Cao
- 3,018
4
votes
5 answers
4
votes
1 answer
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 $$ \int_{\mathbb{R}}\left( \frac{\sin x}{x}\right )^2 $$ using that…
operatorerror
- 29,991
3
votes
1 answer
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 \\ \sin(x) & 0 \leq x < \pi \\ \end{cases} \end{equation} which, in an…
Merkel_Bot
- 103
3
votes
2 answers
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 should I approach it?
3
votes
0 answers
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 Identity. I have discovered that it is the inner product…
user10478
- 2,238
3
votes
3 answers
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 exercise is to use Parseval's identity. Attempt:…
TOMILO87
- 540
3
votes
1 answer
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 $x_{2}^{*}(t)=t \exp(-t^2)$ . I don't know how to proceed…
why_me
- 435
3
votes
1 answer
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 $$f(x)=\left\{\begin{array}{ll}1, & x \in] a, b[ \\ 0, & x \in]-\pi, \pi[\backslash[a, b]\end{array}\right.$$ It is shown that the (complex)…
mhj
- 177
3
votes
1 answer
$L^2$ 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. $\lim_{n \to \infty} <R_n(x),R_n(x)>$ $=0$ $\iff$ $\sum_{n=1}^{\infty}c_i^2 = \int_{-\pi}^{\pi}(f(x))^2dx$ (Parseval identity where $c_i$ s are Fourier…
user519955
- 1,335
3
votes
2 answers
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 imagine how the two integrals can be equal. If we…
Weigh
- 95