Does anyone have recommendations for a good book which explains how Fourier analysis is applied in engineering and physics that does not assume any knowledge of the above topics? I'd like to see the basics of how Fourier analysis is applied to signals, image processing, etc. but I don't know the first thing about the required physics.
This book: http://www.abdn.ac.uk/~mth192/html/music.pdf is free and explains the relationship between mathematics in music. To properly do that you need Fourier series, and so he covers them and then proceeds to show how they apply to music.
In addition to the wonderful book by Dym and McKean recommended by Michael Hardy above, no such list would be complete without recommending the terrific books of Tom Korner: Fourier Analysis and Exercises In Fourier Analysis. Assuming only a rigorous course in calculus a la Spivak or Ross, Korner covers the basic theory including a host of applications such as a very complete coverage of the harmonic occillator and its many variations in potential theory, the heat equation, Brownian motion and stochastic analysis. These are must-have books for anyone interested in harmonic analysis, in either mathematics or the physical sciences.