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 ∅(x) and γ(x) = ∅(1/x)/x have absolutely convergent integrals over the positive real line. Let F(x) = Σ(∞)/(n=1)∅(n/x)/x - ∫(∞)/0∅(t)dt and G(x) = Σ(∞)/(n=1)γ(n/x)/x - ∫(∞)/0γ(t)dt. Then F(x) and G(x) are a Fourier cosine transform pair. We term F(x) the 'discrepancy' of ∅ because it is the error in estimating the integral of ∅ by its Riemann sum with the constant mesh spacing 1/x.
|Original language||English (US)|
|Number of pages||3|
|Journal||Proceedings of the National Academy of Sciences of the United States of America|
|State||Published - 1991|
- Euler-Maclaurin sum formula
- Fourier transforms
- Moebius series