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Self-Fourier functions and self-Fourier operators
Theodoros P Horikis1, Matthew S McCallum
1Engineering Sciences and Applied Mathematics, McCormick School of Engineering, Northwestern University, Evanston, Illinois 60208-3125, USA. theodoros@northwestern.edu
Summary
Researchers discovered infinite families of self-Fourier functions, which equal their Fourier transform. An algorithm was developed for generating and classifying these functions, simplifying complex integral evaluations.
Area of Science:
- Mathematical Physics
- Harmonic Analysis
- Signal Processing
Background:
- Self-Fourier functions, equaling their Fourier transform, are typically limited to known examples like Gaussian and Dirac delta comb functions.
- The mathematical characterization and generation of such functions are often constrained by complex integral computations.
Purpose of the Study:
- To demonstrate the existence of an infinite number of distinct families of self-Fourier functions.
- To provide a novel algorithm for generating and classifying these function families.
- To offer a formalism that bypasses challenging integral evaluations.
Main Methods:
- Development of a new algorithmic approach for function generation and classification.
- Theoretical framework to identify and characterize self-Fourier function families.
- Formalism designed to avoid direct computation of Fourier or transform-type integrals.
Main Results:
- Proof of the existence of infinitely many distinct families of self-Fourier functions.
- A systematic algorithm for generating and categorizing these functions.
- A method that circumvents the need for difficult integral calculations.
Conclusions:
- The study expands the known set of self-Fourier functions beyond traditional examples.
- The developed algorithm offers a simplified and generalizable method for studying these functions.
- This approach provides new theoretical insights into functions with self-Fourier properties.