Transform Theory Reference Request 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).

If you would like to study Laplace transform in depth, there is no better book than Joel Schiff's book on Laplace Transform: http://www.springer.com/gp/book/9780387986982 
If you would like to study a large variety of transforms and see how they relate to each other I highly recommend 
Advanced Engineering Mathematics with MATLAB, Third Edition By Dean G. Duffy
https://www.amazon.ca/Advanced-Engineering-Mathematics-MATLAB-Third/dp/1439816247
In this book they talk about Hilbert Transform, Z-transform, Fourier, Laplace, everything you can think of.
