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]