I have a question regarding the fourier and laplace transform.
First, the Fourier transform essentially takes a function, divides it by a frequency (imaginary exponential), and then sees how much of a frequency exists within the original signal. It does this for (theoretically) every signal thus providing us with information in the frequency domain.
The Laplace transform replaces the frequency variable with the s variable. The s variable is made up of a real and imaginary component. So the Laplace transform essentially takes a function, divides by a real exponential, and then does the Fourier transform on the new signal. Doing this for every value of s then produces a 2D plane.
However, I'm curious if there exists a transformation that simply takes a function, and divides by solely by real exponentials. Or more simply put, I'm asking if there exists something similar to the Fourier Transform but instead of dividing by an imaginary exponential, dividing by a real exponential. The Laplace function somewhat does this, but it divides by both a real exponential and imaginary exponential.
I apologize if I am wording this poorly. Please ask if you need clearing up of anything.