I'm stuck on this concept...
How does one use a Fourier Transform to evaluate an integral?
Thank you!
In a PM the OP stated the question as:
Consdider the function:Well, here's the question. (I hope that you can be of assistance!)
Let f(t) be the even function that is zero if t > 1, and is given by the straight line 1 - t on [0,1]. Find the Fourier transform of f(t), and use it to evaluate the integrals from zero to infinity for (sincx)^2 and (sincx)^4. Comment on the possibility of integrating other powers of sincx.
Thank you so much!
$\displaystyle g(x)=\begin{cases}
1, & x \in [-1/2,1/2]\\
0, & \text{otherwise}
\end{cases}$
Find the FT of $\displaystyle g(x)$ then use the convolution theorem to find the FT of $\displaystyle (g*g)(x)=f(x)$ (check that I have not lost any constants in this)
The FT of $\displaystyle g*g$ is $\displaystyle H(\omega)=G(\omega).G(\omega)$ which will be related to $\displaystyle [\text{sinc}(\omega)]^2$ if you have done all the algebra correctly.
Now evaluate the IFT $\displaystyle H$ at $\displaystyle x=0$ and you should have your result.
In hand waving terms: the FT of a top-hat function is a sinc, and the convolution of a top-hat with itself is a triangular function and the FT of that is therefore the square of a sinc. The inverse FT evaluated at x=0 (or t=0 depending on the convention for variable naming you use) is the integral of the function and so you can evaluate the the integral of the square of the sinc by keeping track of all the multipliers that arrise in the FT and IFT process.
CB
CB