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My question for this thread concerns the application of (mainly) mathematical analysis to fields such as signal processing and machine learning. More specifically, I was wondering if you happen to know of some interesting application of things like measure theory or functional analysis in these fields, that might have the potential to be made into a research questions?

My reason for asking is that I'm an electrical engineering student who is looking around for possible research questions for my Master's thesis, and I’m currently suffering from a lack of inspiration. I have a slightly more theoretical interest than most engineering students and I have also taken more math courses than most engineering students, so I thought it would be fun if I could apply the things I have learned in mathematics to something that is also relevant for "information processing" of some kind (e.g. signal processing, machine learning, and the like, which is related to my actual major/specialisation). My mathematical background is not great, but I have taken a basic course in analysis (on the level of Baby Rudin) and also a more advanced course in analysis, treating measure theory and functional analysis (on the level of "Foundations of Modern Analysis" by Avner Friedman). I have also taken courses in abstract algebra, probability and a rigorous course in complex analysis if that's interesting.

Sorry for a vague and very open-ended question. As I hinted at, I am more or less just looking for inspiration.