Dualizing the Poisson summation formula.
If f(x) and g(x) are a Fourier cosine transform pair, then the Poisson summation formula can be written as 2sumfrominfinityn = 1g(n) + g(0) = 2sumfrominfinityn = 1f(n) + f(0). The concepts of linear transformation theory lead to the following dual of this classical relation. Let phi(x) and gamma(x) = phi(1/x)/x have absolutely convergent integrals over the positive real line. Let F(x) = sumfrominfinityn = 1phi(n/x)/x - integralinfinity0phi(t)dt and G(x) = sumfrominfinityn = 1gamma (n/x)/x - integralinfinity0 gamma(t)dt. Then F(x) and G(x) are a Fourier cosine transform pair. We term F(x) the "discrepancy" of phi because it is the error in estimating the integral phi of by its Riemann sum with the constant mesh spacing 1/x.
Duffin
R J
RJ
Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA.
Weinberger
H F
HF