That's why the Borweins call it the "first bite" theorem! Great minds think alike. Bob --- Eugene Salamin wrote:
From: Warren Smith <warren.wds@gmail.com> To: math-fun@mailman.xmission.com Sent: Saturday, February 25, 2012 4:05 PM Subject: [math-fun] Mathematical hell
I just saw this wikipedia article: http://en.wikipedia.org/wiki/Borwein_integral
Truly, the Gods are against us.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
Let sinc(x) = sin(πx)/(πx). Let rect(x) = 1 for |x| < 1/2, otherwise 0. Define Fourier transform to have kernel exp(+-2πitx). Then rect(t) and sinc(x) are a Fourier transform pair. The convolution of two functions f and g is defined as (f*g)(x) = int(f(x-y) g(y), y=-inf..inf) = = int(f(y) g(x-y), y=-inf..inf). With F denoting the Fourier transform operation, the convolution theorem states that F(f*g) = F(f) F(g), F(fg) = F(f)*F(g). This shows that convolution is commutative and associative. Also, under scaling, let g(x) = f(x/a). Then F(g)(t) = |a| F(f)(at). In particular, for a > 0, sinc(x/a) and (a rect(ax)) are Fourier transform pairs.
We have 1 = rect(0) = F(sinc)(0) = int(sinc(x), x=-inf..inf). Next, taking a to be positive,
int(sinc(x) sinc(x/a), x=-inf..inf) = F[sinc(x) sinc(x/a)](t=0) =[[rect(t)] * [a rect(at)]](t=0).
In taking the convolution the rect(at) nibbles away at each of the edges of rect(t) by amount a/2. As long as a < 1. the nibbling doesn't reach t = 0, and the integral equals 1. Similarly,
int(sinc(x) sinc(x/a) sinc(x/b) sinc(x/c) ... , x=-inf..inf) = [[rect(t)] * [a rect(at)] * [b rect(bt)] * [c rect(ct)] * ...](t=0).
Again the nibbles fail to reach 0, exactly when a + b + c + ... < 1. By continuity, we can say that the integral equals 1 when a + b + c + ... <= 1. But if the sum is greater than 1, the convolutions will have eaten their way to the center, and the integral is less than 1.
The same trick applies to any band-limited function, one whose Fourier transform has compact support. For example BesselJ(0,x) and 1/sqrt(1 - t^2) for |t| < 1, other wise 0, are a Fourier transform pair (modulo various factors of π).
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