I am working on my understanding of various transforms and I have been thinking about the Fourier transform, what i "does" to the function it is applied to.
The way I see it:
The function $f$ that is transformed is multiplied with $\exp(icx)$ which essentially describes a rotating vector in the complex plane.
If $f$ is periodic cosine and the period of $f$ does not match the period of $\exp(icx)$ the "terms" in the integral will vary and cancel each other leading to a value of zero for the transform.
If $f$ is periodic cosine and the period of $f$ matches the period of $\exp(icx)$ the "terms" in the integral will be constant and the value of the transform will be $\infty$.
If $f$ is periodic but not a cosine it can be decomposed to a sum of cosines and the different cosine terms of this sum will work as above resulting in a spectra for $f$.
If this is somewhat correct I wonder:
What about aperiodic functions?
Is there a similar way of thinking about the Laplace transform?
Please forgive the non mathematical language, I'm neither a math major nor is English my mother tongue.