Mathematics behind Signal and Systems

[amg63]

I am looking for a signal processing textbook that uses real, complex, and functional analysis with measure theory. In other words, mathematically rigorous signal processing. Specifically, I prefer the kind that takes time to review all the topics from mathematical analysis before jumping into the fire (due to a potential experience deficit in some analysis topics), but will gladly consider any rigorous signal processing. A Mathematical introduction to signal and systems by A. D. Lewis is pretty much what I am looking for, however, only first volume is available, so I need an equivalent. Please inform me of any textbooks that you know of that fit my search criteria, and I shall be grateful in return. Thank you for your time.

 

[Baluncore]

You may be interested in the “Digital Signal Processing” ebook produced by the CERN Accelerator School.
Available as .pdf from; http://cds.cern.ch/record/1003726?ln=en

 

[osilmag]

Google the Fourier Transform and become an expert at that.

 

[deskswirl]

I think this is the book you are looking for: http://www.fourierandwavelets.org/FSP_v1.1_2014.pdf

EDIT: After looking at the text you posted perhaps my suggestion is not rigorous enough.

Note to moderators this version of the book is posted for free distribution on the author's website.

 

[jasonRF]

I wasn't familiar with the notes by Lewis, but was intrigued so googled. I found the second volume on the author's web site.
http://www.mast.queensu.ca/~andrew/
Follow the link to course notes, then math 334.

One book I really like is "Fourier Analysis and its applications" by Gasquet and Witomski. It is probably not advanced enough for you; it is certainly less advanced (and shorter!) than the notes by Lewis, but is more polished. I especially like the treatment of Fourier analysis and distribution theory.

Jason