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Dualizing the Poisson summation formula.

R J Duffin1, H F Weinberger

  • 1Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA.

Proceedings of the National Academy of Sciences of the United States of America
|August 15, 1991
PubMed
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This study introduces a dual relationship to the Poisson summation formula using linear transformation theory. It defines a "discrepancy" function related to Riemann sums and Fourier cosine transforms.

Area of Science:

  • Mathematical Analysis
  • Harmonic Analysis

Background:

  • The Poisson summation formula relates sums of function values to integrals.
  • Fourier cosine transforms are fundamental in signal processing and physics.

Purpose of the Study:

  • To establish a dual relationship to the classical Poisson summation formula.
  • To introduce and analyze the concept of a "discrepancy" function.

Main Methods:

  • Utilizing concepts from linear transformation theory.
  • Defining new functions F(x) and G(x) based on a function and its reciprocal transformation.
  • Establishing F(x) and G(x) as a Fourier cosine transform pair.

Main Results:

  • A novel dual relation to the Poisson summation formula is derived.

Related Experiment Videos

  • The function F(x) is identified as the "discrepancy"—the error in Riemann sum estimation.
  • The derived functions F(x) and G(x) form a Fourier cosine transform pair.

    • Conclusions:

    • The study extends classical summation formulas using Fourier transforms.
    • The introduced "discrepancy" provides insight into numerical integration errors.
    • This work offers a new perspective on the interplay between discrete sums and continuous integrals.