I am not a Mathematician - am just a software developer though I did some "Math" back in the day as part of my undergrad studies millions of years ago.

Recently I had to revisit Fourier analysis of signals for an application I am working on, and it struck me that if you squinted at it just right, it had some similarity to the representations of numbers in an integer base as given here (with the trigonometric or complex exponential functions being replaced by powers of the base integer, e.g., $123 = 1 \times 10^2 + 2 \times 10^1 + 3 \times 10^0$). I understand that the powers of integers are not orthogonal to each other, but otherwise, I was wondering if there is any overarching "theory" that includes these two as special cases? Does, for e.g., Harmonic analysis cover representation of integers by powers of integer bases?

Edit: the same I suppose could be said for the Taylor series expansion of functions as well, with the harmonic exponential or trigonometric functions being replaced by powers of variables.

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  • The Taylor series - Fourier transform relation is explained here: http://math.stackexchange.com/questions/7301/connection-between-fourier-transform-and-taylor-series , as I later found out. The basis representation question still remains. – ARV Dec 26 '15 at 12:11
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    In fact, power series (Taylor series, for example) are very closely related to Fourier series. The $Z$-transform makes the connections more explicit. – Ben Grossmann Dec 26 '15 at 12:15
  • I don't know about basis representations of numbers, though. If you try and frame it in the linear algebraic sense of a "basis", then a lot of things don't quite fit. – Ben Grossmann Dec 26 '15 at 12:20
  • @Omnomnomnom Thanks for your reply. I have attempted to make the question clearer. By "basis representation", I meant representation of a number in a given base. E.g., $123 = 1 \times 10^2 + 2 \times 10^1 + 3 \times 10^0$ – ARV Dec 26 '15 at 12:42
  • I understand, but the statement stands – Ben Grossmann Dec 26 '15 at 14:02

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