Fractional Fourier Series on the Torus and Applications

This paper introduces the fractional Fourier series on the fractional torus and proceeds to investigate several fundamental aspects. Our study includes topics such as fractional convolution, fractional approximation, fractional Fourier inversion, and the Poisson summation formula. We also explore th...

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Published inFractal and fractional Vol. 8; no. 8; p. 494
Main Authors Wang, Chen, Hou, Xianming, Wu, Qingyan, Dang, Pei, Fu, Zunwei
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.08.2024
Subjects
Approximation
Boundary conditions
Convergence
Fourier series
Fourier transforms
fractional approximate identity
fractional Fejér kernel
fractional Fourier inversion
fractional Fourier series
Partial differential equations
Signal processing
Smoothness
Toruses
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Summary:This paper introduces the fractional Fourier series on the fractional torus and proceeds to investigate several fundamental aspects. Our study includes topics such as fractional convolution, fractional approximation, fractional Fourier inversion, and the Poisson summation formula. We also explore the relationship between the decay of its fractional Fourier coefficients and the smoothness of a function. Additionally, we establish the convergence of the fractional Féjer means and Bochner–Riesz means. Finally, we demonstrate the practical applications of the fractional Fourier series, particularly in the context of solving fractional partial differential equations with periodic boundary conditions, and showcase the utility of approximation methods on the fractional torus for recovering non-stationary signals.
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ISSN:2504-3110
2504-3110
DOI:10.3390/fractalfract8080494