## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 7348-7350 |

Number of pages | 3 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 88 |

Issue number | 16 |

State | Published - 1991 |

## Keywords

- Euler-maclaurin sum formula
- Fourier transforms
- Moebius series