Search All of the Math Forum:

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

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

 Messages: [ Previous | Next ]
 jhunpingco@my-dejanews.com Posts: 25 Registered: 12/7/04
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

-----== Posted via Deja News, The Leader in Internet Discussion ==-----
http://www.dejanews.com/rg_mkgrp.xp Create Your Own Free Member Forum

Date Subject Author
9/19/98 jhunpingco@my-dejanews.com
9/19/98 Paul Hughett
9/19/98 David Kastrup
9/19/98 Ronald Bruck
9/20/98 David C. Ullrich
9/20/98 David C. Ullrich
9/28/98 DGoncz