From: phunt@interpac.net
Subject: Re: Fourier's Theorem?
Date: Sun, 17 Oct 1999 17:34:31 GMT
Newsgroups: sci.math
Keywords: Statement of Fourier's theorem (representation by series)
On 17 Oct 1999 12:18:35 -0400, lrudolph@panix.com (Lee Rudolph) wrote:
>....
>Anyway, try this search for a start; then, if you can't find a decent
>statement among the ramblings and babblings of the hemidemisemi-educated
>musicians and whacked-out New Age crystallographers, get back to us
>and maybe someone will crack a book.
>
>http://www.altavista.com/cgi-bin/query?pg=aq&w=w&text=y&q=fourier's+theorem
>
>Lee Rudolph
>
Okay Lee, I broke down and looked it up in
my old-school text:
"Advanced Mathematics for Engineers, 2nd Ed."
by H.W. Reddick & F.H. Miller. New York:
John Wiley & Sons; London: Chapman & Hall,
Ltd., 1938-1947. Tenth printing 1953.
Page 185:
Theorem II.
Any single-valued function f(x), continuous
except possibly for a finite number of finite
discontinuities in an interval of length 2*pi,
and having only a finite number of maxima and
minima in this interval, possesses a convergent
Fourier series representing it.
That's it ... That's all they wrote.
/ph
- - - - - - -
>"P Warren" writes:
>
>>Hi there,
>>am 17, in final year of High School, doing top maths, and need help.
>>
>>I have been given a question to research as part of a 'history of calculus'
>>project.
>>
>>"Look at fourier's theorem, elaborate on this theorem, show worked examples
>>of problems using fouriers theorem."
>>
>>Only problem is, i can't find ANYWHERE, where fourier's theorem is actually
>>stated, either as words, or in symbols. Does it really exist?
>
>A websearch at AltaVista turns up 104 hits (many clearly of dubious
>value), so I'm rather surprised that you can't find it stated ANYWHERE.
>Anyway, try this search for a start; then, if you can't find a decent
>statement among the ramblings and babblings of the hemidemisemi-educated
>musicians and whacked-out New Age crystallographers, get back to us
>and maybe someone will crack a book.
>
>http://www.altavista.com/cgi-bin/query?pg=aq&w=w&text=y&q=fourier's+theorem
>
>Lee Rudolph
>
==============================================================================
From: "David C. Ullrich"
Subject: Re: Fourier's Theorem?
Date: Sun, 17 Oct 1999 13:03:32 -0500
Newsgroups: sci.math
phunt@interpac.net wrote:
[previous article quoted, through: --djr]
> That's it ... That's all they wrote.
I believe that this is what people mean when they say "Fourier's
Theorem" - this or related results saying that under certain conditions
a function has a convergent Fourier series.
Calling it "Fourier's Theorem" isn't really quite right. What Fourier
said was something to the effect that any function whatever has a
convergent Fourier series - he could never quite prove this, which
turned out to be understandable, since it's not true. Not true
even for continuous functions. Finding extra hypotheses that
would give convergence led to a lot of analysis (I think that
the version of "Fourier's theorem" above is actually due to
Dirichlet, for example - don't quote me on that.)
[rest of quoted article deleted --djr]