I've come across laplace transforms as a method to solve differential/integral equations by diagonalizing the derivative operator. I've also seen generating functions as a similar way to transform a recurrence relation into an algebraic problem by diagonalizing the shift operator. The laplace transform of a convolution is a product of the laplace transforms for each function and similarly the generating function of a convolution is a product of generating function. A different application of transforms would be how the dirichlet convolution can become a product when using Dirichlet generating functions. Yesterday I was reading about multiplication algorithms and discovered one fast way to multiply numbers is based on doing a discrete fourier transform because a product of two numbers can be viewed as a convolution of two polynomials where the x in the polynomials would be the base you use to represent the numbers.
These examples of using transforms to simplify a problem I've seen from various courses in math, but I've never really studied the transforms for there own sake. What would be a good textbook/other source on transform theory that deals with things like the similarities of various integral transforms and generating functions?
Edit: My current math background is essentially a standard undergraduate math courses (introductory algebra, analysis, topology, discrete math, etc).