
A subtle and interesting "contradiction" in Fourier analysis
Posted:
Sep 19, 1998 1:18 PM


Hi,
Here is an interesting question:
Is the sum of two periodic functions also a periodic function?
For example, if x1(t) and x2(t) are both periodic, each with respective periods T1 and T2, and s(t) = x1(t) + x2(t), is s(t) also periodic?
In general, the answer is no. s(t) is only periodic so long as the T1/T2 is a rational number. If it is an irrational number, then s(t) is not periodic.
Note that in this case, s(t) is not periodic and not an absolutely integrable function, which means that its Fourier transform doesn't exist in the usual sense.
Does this little illustration show a hole in the usual Fourier analysis? Given any band of frequencies, I can certainly construct a frequency in that band like s(t) as defined above for which the Fourier transform does not exist and for which I am stalled in my analysis.
Have I just confused myself, or does this "hole" really exist?
Thanks.
jose
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