On Eigenfunctions of the Fourier Transform
- Авторлар: Lanzara F.1, Maz’ya V.2,3
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Мекемелер:
- Sapienza University of Rome
- University of Linköping
- University of Liverpool
- Шығарылым: Том 235, № 2 (2018)
- Беттер: 182-198
- Бөлім: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/242086
- DOI: https://doi.org/10.1007/s10958-018-4067-7
- ID: 242086
Дәйексөз келтіру
Аннотация
A nontrivial example of an eigenfunction in the sense of the theory of distributions for the planar Fourier transform was described by the authors in their previous work. In this paper, a method for obtaining other eigenfunctions is proposed. Positive homogeneous distributions in ℝn of order −n/2 are considered, and it is shown that F(ω)|x|−n/2, |ω| = 1, is an eigenfunction in the sense of the theory of distributions of the Fourier transform if and only if F(ω) is an eigenfunction of a certain singular integral operator on the unit sphere of ℝn. Since \( {Y}_{m,n}^{(k)}\left(\omega \right){\left|\mathbf{x}\right|}^{-n/2} \), where \( {Y}_{m,n}^{(k)} \) denote the spherical functions of order m in ℝn, are eigenfunctions of the Fourier transform, it follows that \( {Y}_{m,n}^{(k)} \) are eigenfunctions of the above-mentioned singular integral operator. In the planar case, all eigenfunctions of the Fourier transform of the form F(ω)|x|−1 are described by means of the Fourier coefficients of F(ω).
Авторлар туралы
F. Lanzara
Sapienza University of Rome
Email: vladimir.mazya@liu.se
Италия, 2, Piazzale Aldo Moro, Rome, 00185
V. Maz’ya
University of Linköping; University of Liverpool
Хат алмасуға жауапты Автор.
Email: vladimir.mazya@liu.se
Швеция, Linköping, 581 83; Liverpool, L69 3BX