The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: A subtle and interesting "contradiction" in Fourier analysis
Replies: 6   Last Post: Sep 28, 1998 7:59 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 25
Registered: 12/7/04
A subtle and interesting "contradiction" in Fourier analysis
Posted: Sep 19, 1998 1:18 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply


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

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

Have I just confused myself, or does this "hole" really exist?



-----== Posted via Deja News, The Leader in Internet Discussion ==----- Create Your Own Free Member Forum

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.