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I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without mention of convergence or function spaces), the Laplace and Fourier transforms, and the sampling theorem.

I still don't truly understand the sifting property, or why sampling is multiplication with a shifted delta instead of a shifted 1 (which would preserve the value of the signal at that point right?). I'm still not sure how to determine when a signal's Fourier series will converge, when you are allowed to take a Fourier or Laplace transform, or whether the derivations of the transform properties we learned are mathematically justified and rigorous.

This led me to self study books like Tolstov's Fourier Series and Richards's Theory of Distributions, which I could not follow due to my non-rigorous math background. The Osgood Stanford lectures were quite helpful, but I still feel quite sad that I do not have a deeper understanding of my math tools.

According to my school's EE graduate course catalog, there are a couple of classes covering Hilbert, Banach, and L^p spaces, and linear functionals. Is it possible to cover these topics without a formal mathematical background? Should I begin self learning real and complex analysis to prepare for these topics?

Forgive me for this lengthy question. Thanks!

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    $\begingroup$ Complex analysis would indeed be helpful; it's a major tool used in EE (Or so I am told...); buying a crisp copy of Rudin's 'Real and Complex Analysis' would be great. Do check out the prerequisites though. $\endgroup$
    – Panglossian Oporopolist
    Jul 1, 2015 at 2:04
  • $\begingroup$ Physics and engineering students are often exposed heuristically to areas of analysis to facilitate their applications. That said, there are physics and engineering professors who do teach their students using a mathematically rigorous approach - even at the undergraduate level. And students in applied area are highly encouraged to enroll in math courses that naturally incorporate rigor. Self learning is also always encouraged! $\endgroup$
    – Mark Viola
    Jul 1, 2015 at 2:24
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    $\begingroup$ @BooBruEEn Here's a pdf version of the aforementioned: ruangbacafmipa.staff.ub.ac.id/files/2012/02/… Here's the pdf of the book which should be vanquished before the above book; Rudin's Principle of Math. Analysis: notendur.hi.is/vae11/%C3%9Eekking/… (Of course, you might have already finished some things in this book). $\endgroup$
    – Panglossian Oporopolist
    Jul 1, 2015 at 2:30
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    $\begingroup$ I wouldn't recommend Rudin's Real and Complex Analysis for most electrical engineering PhDs. That book seems like overkill. I think there are easier books that cover the relevant theory. $\endgroup$
    – littleO
    Jul 1, 2015 at 3:21
  • $\begingroup$ It really depends on what area of research you are thinking of. A course in real analysis & integration (Lebesgue) would be useful for the latter topics. I still find Rudin hard going for new topics, so I would not recommend it for self study. Personally I learn better & faster when I have people I can discuss topics with, so taking classes may be better in that respect. $\endgroup$
    – copper.hat
    Jun 25, 2020 at 22:25

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$Q1.$ According to my school's EE graduate course catalog, there are a couple of classes covering Hilbert, Banach and L^p spaces; and linear functionals. Is it possible to cover these topics without a formal mathematical background?

$A1.$ It is the responsibility of the school to provide a curriculum for the students that is both meaningful an doable. Therefore if the graduate courses have classes on some advanced mathematics subjects, then it is reasonable to assume that the undergraduate mathematics curriculum of the school provides the necessary and required background. As far I can see there is no particular reason why you, as an undergraduate student, have to worry about these things. But when you are in doubt you can always ask your study adviser about this.

$Q2.$ Should I begin self learning real and complex analysis to prepare for these topics?

$A2.$ No, I don't think that is necessary.

$Q3.$ Do electrical engineering researchers usually know higher mathematics? For example measure and distribution theory, or functional analysis.

$A3.$ No they don't. They will typically have excellent experimental and theoretical knowledge of analog (electric) and digital (electronic) signal processing. They know physics and mathematics and computer science, but not at the highest level.

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