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 2Σ_n=1^∞g(n) + g(0) = 2Σ_n=1^∞f(n) + f(0). The concepts of linear transformation theory lead to the following dual of this classical relation. Let φ(x) and γ(x) = φ(1/x)/x have absolutely convergent integrals over the positive real line. Let F(x) = Σ_n=1^∞φ(n/x)/x - ∫₀^∞φ(t)dt and G(x) = Σ_n=1^∞γ(n/x)/x - ∫₀^∞γ(t)dt. Then F(x) and G(x) are a Fourier cosine transform pair. We term F(x) the "discrepancy" of φ because it is the error in estimating the integral of φ by its Riemann sum with the constant mesh spacing 1/x.