Question: What is the difference between and both found in Wikipedia?
Answer: Substantially nothing. Many/most EE texts use the 1st one to represent a Fourier transform pair. Mathematicians insist on the 2nd one. My favorite textbook author, Van Trees (Van Trees, Harry L (1968). Detection, Estimation, and Modulation Theory. Vol. 1. New York: John Wiley. p. 680. ISBN0-471-09517-6.) uses
My comments:
is a universally recognized symbol for abscissa (independent variable)... the argument of a function. So I agree with the mathematicians on that count.
But I agree with the EEs on reserving for frequency (in cycles per unit of time or space).
I tried (and failed) to convince the mathematicians to substitute the less intimidating instead of .
I reluctantly agree that the operator notation is more consistent in some applications; e.g. or instead of switching from cap letters to and but that seems like a minor consideration to me.
Bottom line: I like (for "signal") and (for "spectrum") instead of and or and . I also prefer instead of
is N-periodic in n, without assuming S[k] is periodic. Following the example of the continuous time Fourier series, it seems that any coefficient S[m] can be computed from one period of as follows: