I'm found of seeing the Fourier transform as a fancy way of changing basis / coordinates, not sure how mathematically correct is that.
In high school physics some problems got way easier by redefining coordinates as x' and y', solving for those, them going back to x and y. That's what the Fourier transform does, but for functions.
Looking at its formula, we can see it looks like we are projecting a function into a series of complex exponentials, just like we need to project a vector in x' and y' for a change of basis in the euclidian space.
Then, the Fourier coefficients can be thought as "how much my function looks like a complex exponential of this given frequency". Compare the formulas for the Pearson Correlation and Convolution, they are almost the same.
Since the Fourier transform is just a particular case of changing basis, there are many, many other integral transforms that might be useful in different contexts. We can even create our own. But unis teach about the Fourier transform because it is really good for solving differential equations, which turns out to be a popular way of modeling problems in engineering.
In high school physics some problems got way easier by redefining coordinates as x' and y', solving for those, them going back to x and y. That's what the Fourier transform does, but for functions.
Looking at its formula, we can see it looks like we are projecting a function into a series of complex exponentials, just like we need to project a vector in x' and y' for a change of basis in the euclidian space.
Then, the Fourier coefficients can be thought as "how much my function looks like a complex exponential of this given frequency". Compare the formulas for the Pearson Correlation and Convolution, they are almost the same.
Since the Fourier transform is just a particular case of changing basis, there are many, many other integral transforms that might be useful in different contexts. We can even create our own. But unis teach about the Fourier transform because it is really good for solving differential equations, which turns out to be a popular way of modeling problems in engineering.