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Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Fourier uniqueness in even dimensions.

Andrew Bakan1, Haakan Hedenmalm2, Alfonso Montes-Rodríguez3

  • 1Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv 01601, Ukraine.

Proceedings of the National Academy of Sciences of the United States of America
|April 8, 2021
PubMed
Summary
This summary is machine-generated.

This study presents a new method for Fourier uniqueness, enabling unique function reconstruction from discrete data. This approach, based on Klein-Gordon equation theory, offers an alternative to modular forms for even dimensions.

Keywords:
Fourier transformFourier uniquenessHeisenberg uniqueness pairsKlein–Gordon equation

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Area of Science:

  • Harmonic Analysis
  • Number Theory
  • Partial Differential Equations

Background:

  • Previous work utilized modular forms for Fourier uniqueness results in specific dimensions.
  • These results had significant applications, including solving the sphere packing problem in dimensions 8 and 24.

Purpose of the Study:

  • To present an alternative approach to Fourier uniqueness results using the Klein-Gordon equation.
  • To explore the connection between Klein-Gordon uniqueness theory and modular forms.

Main Methods:

  • The study employs the uniqueness theory for the Klein-Gordon equation.
  • This theory involves the study of iterations of Gauss-type maps.

Main Results:

  • An alternative method for achieving Fourier uniqueness in even dimensions is presented.
  • The derivation provides conditions for Fourier interpolation, potentially optimal.

Conclusions:

  • The Klein-Gordon equation offers a novel pathway to Fourier uniqueness results.
  • This suggests a link between Gauss-type map iterations and modular forms in harmonic analysis.