Intereting Posts

What are the rules for complex-component vectors and why?
Lang's treatment of product of Radon measures
What should we call the 'sets' which don't exist under certain set theory axioms?
How to find $\int_{0}^{1}\dfrac{\ln^2{x}\ln^2{(1-x)}}{2-x}dx$
Non-revealing maximum
Proving that every patch in a surface $M$ in $R^3$ is proper.
Schwartz impossibility result
Is it possible that $a_n>0$ and $\sum a_n$ converges then $na_n \to 0$? (without assuming $a_n$ is decreasing)
ArcTan(2) a rational multiple of $\pi$?
How to teach mathematical induction?
Probability of $\limsup$ of a sequence of sets (Borel-Cantelli lemma)
What is a curve? (Definition)
Hall's theorem vs Axiom of Choice?
Smallest possible triangle to contain a square
Bourbaki exercise on connected sets

The question at hand is to use Parseval’s theorem to solve the following integral:

$$\int_{-\infty}^{\infty} sinc^4 (kt) dt$$

I understand Parseval’s theorem to be:

- Fourier Transform Proof $ \mathcal F(f(x)g(x))=(\frac{1}{2\pi})F(s)*G(s)$
- How to prove this equality .
- Fourier Transform - Laplace Equation on infinite strip - weird solution involving series
- Fourier Analysis and its applications
- Find the solution of the Dirichlet problem in the half-plane y>0.
- Dirac Delta and Exponential integral

$$E_g = \int_{-\infty}^{\infty} g^2(t) = \int_{-\infty}^{\infty} |G(f)|^2 df $$

I began by doing the obvious and removing the squared such that:

$$g^2(t) = Sinc^4 (kt)$$

$$g(t) = Sinc^2 (kt)$$

Following the table of Fourier transforms in my book, I see that

$B*sinc^2({\pi}Bt)$ has the transform $\Delta(\frac{f}{2B})$. However, I’m stuck at this becuase I’m not sure how I can integrate the $\Delta$. I also feel as though I am overthinking this problem – any assistance would be greatly appreciated! Thank you so much in advanced!

- Help with $\int \frac 1{\sqrt{a^2 - x^2}} \mathrm dx$
- How to find $\lim_{n \to \infty} \int_0^1 \cdots \int_0^1 \sqrt{x_1+\sqrt{x_2+\sqrt{\dots+\sqrt{x_n}}}}dx_1 dx_2\dots dx_n$
- Comparison between integrals
- Integrate $e^{ax}\sin(bx)?$
- Integral $\int_0^{\pi/2} x\cot(x)dx$, Differntiation wrt parameter only.
- Integral help here please?
- Simplifying the integral $\int\frac{dx}{(3 + 2\sin x - \cos x)}$ by an easy approach
- How to find $\int_{-1}^1 \frac{\cos x}{a^x+1}\mathrm dx$
- Prove that $F(xy) =F(x) + F(y)$ when $F(x)$ is not $\ln(x)$
- Let $f:\to\mathbb{R}$ be a continuous function. Calculate $\lim\limits_{c\to 0^+} \int_{ca}^{cb}\frac{f(x)}{x}\,dx$

Using Parseval’s Theorem, we have

$$\begin{align}

\int_{-\infty}^\infty \text{sinc}^4(kt)\,dt&=\frac{1}{k}\int_{-\infty}^\infty \text{sinc}^4(t)\,dt\\\\

&=\frac{1}{2\pi k}\int_{-\infty}^\infty \left|\mathscr{F}\left(\text{sinc}^2\right)(\omega)\right|^2\,d\omega

\end{align}$$

where

$$\begin{align}

\mathscr{F}\left(\text{sinc}^2\right)(\omega)&=\int_{-\infty}^\infty \text{sinc}^2(t)e^{i\omega t}\,dt\\\\

&=\frac{\pi }{4}\left(|\omega -2|-2|\omega|+|\omega+2|\right)

\end{align}$$

Therefore,

$$\begin{align}

\int_{-\infty}^\infty \text{sinc}^4(kt)\,dt&=\frac{1}{2\pi k} \int_{-\infty}^\infty \left(\frac{\pi }{4}\left(|\omega -2|-2|\omega|+|\omega+2|\right)\right)^2 \,d\omega\\\\

&=\frac{\pi}{16 k}\int_0^2 \left(4-2\omega \right)^2 \,d\omega\\\\

&=\frac{2\pi}{3k}

\end{align}$$

Another approach, maybe easier.

$$ I(k)=\int_\mathbb{R}\text{sinc}(kt)^4\,dt = \frac{1}{k}\int_{-\infty}^{+\infty}\frac{\sin(x)^4}{x^4}\,dx \tag{1}$$

but $\sin(x)^4 = \frac{3}{8}-\frac{1}{2}\cos(2x)+\frac{1}{8}\cos(4x)$ by De Moivre’s formula, so by applying integration by parts three times:

$$ I(k) = \frac{1}{6k}\int_{-\infty}^{+\infty}\frac{\frac{d^3}{dx^3}\sin(x)^4}{x}\,dx = \frac{1}{6k}\int_{-\infty}^{+\infty}\frac{8\sin(4x)-4\sin(2x)}{x}\,dx\tag{2}$$

and:

$$ I(k) = \frac{(8-4)\pi}{6k} = \color{red}{\frac{2\pi}{3k}}\tag{3}$$

follows.

- Variational formulation of Robin boundary value problem for Poisson equation in finite element methods
- If $z$ is the unique element such that $uzu=u$, why is $z=u^{-1}$?
- Concise description of why rotation quaternions use half the angle
- How does Maurer-Cartan form work
- Multiplying three factorials with three binomials in polynomial identity
- Is $\mathbb{Q}$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$?
- Spectrum of infinite product of rings
- Discrete Maths Logically Equivalent
- Numerical computation of the Rayleigh-Lamb curves
- Hillary Clinton's Iowa Caucus Coin Toss Wins and Bayesian Inference
- Distribution of Stopped Brownian motion at hitting time of another Brownian motion.
- Prove that $6|2n^3+3n^2+n$
- Matrix of Infinite Dimension
- Cylinder in 3D from five points?
- Irrationality of $\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$