As a part of an exercize in Fourier transforms I came across this integral: \begin{equation} \int_{-\infty}^{\infty}\frac{f(y)}{(x-y)^2+1} \end{equation} I don't know why, but I felt that this has to be a convolution of two functions. Which two - I didn't have a clue.
According to the answer we have to define a function g(x): $g(x) = \frac{1}{1+x^2}$
Then is says "we can now easily identify that:" \begin{equation} \int_{-\infty}^{\infty}\frac{f(y)}{(x-y)^2+1} = f*g(x) \end{equation} where $f*g$ is the convolution of f and g.
Knowing that the definition of a convolution is: \begin{equation} (f*g)(x) = \int_{-\infty}^{\infty}f(x-y)g(y)dy = \int_{-\infty}^{\infty}f(y)g(x-y)dy \end{equation} ...I have 3 questions:
- How can we "easily" see that the first function is a convolution?
- How do you identify a convolution generally?
- What's the method you experienced people use to identify and define the function with which we convolute f with (like g(x) in this case)?
Thank you!