The Poisson Summation Formula

The Poisson Summation Formula PhL 3.30.11 Overview: This formula is in the context of the regular Fourier Transform. If we define this transform using frequency instead of angular frequency as the variable φ#(ν) ≡ !Syntax Error, I dy φ(y)ei2πνy φ(y) = (1/2π)!Syntax Error, I dν φ#(ν)e-i2πνy Then the Poisson Summation formula says this: Σn=-∞∞φ(n) = Σn=-∞∞ φ#(n) ________________________________________________________________________________ I have about 5 inches of comment on this in the raw Chap 1 notes, but let's do it again here from scratch. (a) It all begins back on page 39 where he considers, in distribution theory, the derivative of a piecewise differentiable function and he shows that f ' = [f '] + Σk Δfk δ(x-ak) where [f '] is the derivative away from the "jumps" which are Δfk located at x = ak. (b) On page 44-45, he applies this idea to the sawtooth function on page 45 and he concludes that, for this function, f '(x) = 1 - Σk=-∞∞ δ(x-k) where [f '] = 1 in this case, and Δfk = 1 as well. So this is one expression for f '(x). But we can get another expression by doing a Fourier Series expansion of the sawtooth which tells us that f(x) = -(2/π) Σn=1∞ sin(2πnx)/(2n) and this is a routine fact that I won't derive in detail, Schaum p 131 or whatever. So we can differentiate this to get our "other" expression for f '(x): f '(x) = -2 Σn=1∞ cos(2πnx) // = 1 - Σk=-∞∞ δ(x-k) We now set our two expressions equal to learn that (old Neumann's factor εn appears here!) Σk=-∞∞ δ(x-k) = 1 + 2 Σn=1∞ cos(2πnx) = Σn=0∞ εn cos(2πnx) 1.22 (c) We now move to page 46 and Example 7. The first action is to rewrite the above using cos(2πnx) = (1/2) (ei2πnx + e-i2πnx) so that Σk=-∞∞ δ(x-k) = 1 + Σn=1∞ (ei2πnx + e-i2πnx) = Σn=-∞∞ ei2πnx p 46 A We next set x = y/λ to get Σk=-∞∞ δ(y/λ-k) = Σn=-∞∞ ei2πny/λ p 46 B = Σk=-∞∞ |λ| δ(y-kλ) Multiply both sides by φ(y) and integrate on y to get !Syntax Error, I dy φ(y) Σk=-∞∞ |λ| δ(y-kλ) = !Syntax Error, I dy φ(y) Σn=-∞∞ ei2πny/λ Σk=-∞∞ |λ|!Syntax Error, I dy φ(y) δ(y-kλ) = Σn=-∞∞ !Syntax Error, I dy φ(y) ei2πny/λ Σk=-∞∞φ(kλ) = (1/|λ|) Σn=-∞∞ Φ(2πn/λ) page 47 1.23b This thing is the Poisson Summation Formula. We have used the Fourier Integral definition !Syntax Error, I dy φ(y)eiky ≡ Φ(k) φ(y) = (1/2π)!Syntax Error, I dk Φ(k)e-iky Using the Volume 2 notation φ^ for a Fourier Integral Transform we can write 1.23b as Σk=-∞∞ φ(kλ) = (1/|λ|) Σn=-∞∞ φ^(2πn/λ) and λ can be an arbitrary number. If we set λ = 1 we get Σk=-∞∞ φ(k) = Σn=-∞∞ φ^(2πn) and this is Vol II page 212 7.47 ! We could alternatively selected λ = 2π to get Σk=-∞∞ φ(k/2π) = (1/2π) Σn=-∞∞ φ^(n) Now in my Spectral Notes page 11 we know there is something called Parseval's Theorem !Syntax Error, I dy | φ(y) |2 = (1/2π) !Syntax Error, I dk | φ^(k) |2 which has a similar look to our series result, but it involves the squares of the functions! There is a Fourier Series version of this theorem that is this: (wiki on Parseval) and there are discrete Fourier transform versions of this as well. For me, the Poisson Summation Formula is completely new. (d) Fourier Integral Transform in Frequency Versus Angular Frequency. Above we had the angular thing like so: !Syntax Error, I dy φ(y)eiky ≡ Φ(k) φ(y) = (1/2π)!Syntax Error, I dk Φ(k)e-iky Now we can write ω = 2πf which here would be k = 2πν where ν is waves per meter say. Then we get !Syntax Error, I dy φ(y)ei2πνy ≡ Φ(2πν) φ(y) = (1/2π)!Syntax Error, I dν Φ(2πν)e-i2πνy We would in this case define a new symbol, let's say φ#(ν) ≡ Φ(2πν) and then we have !Syntax Error, I dy φ(y)ei2πνy ≡ φ#(ν) φ(y) = (1/2π)!Syntax Error, I dν φ#(ν)e-i2πνy and this is our Fourier Integral Transform using the x and ν as the conjugate variables. (e) We had from above Σk=-∞∞φ(kλ) = ( 1/|λ|) Σn=-∞∞ !Syntax Error, I dy φ(y) ei2πny/λ So now we can write !Syntax Error, I dy φ(y) ei2πny/λ = φ#(n/λ) and then our Poisson Sum gizmo says Σk=-∞∞φ(kλ) = ( 1/|λ|) Σn=-∞∞ φ#(n/λ) and now we can set λ = 1 to get Σk=-∞∞φ(k) = Σn=-∞∞ φ#(n) which is certainly a compact looking result! In wiki it appears this way What is this Poisson thing saying? If you add up the values of f(x) at all the integers, you get the same thing you get by adding up f#(ν) at all its integers! This really seems amazing, but I just derived it above following Stak. Now we started with a test function, so you have to make sure that both series converge when you apply something like this.