Arithmetical Fourier transforms and Hilbert space: Restoration of the lost legacy

In this survey-type paper we show that the seemingly unrelated two fields-Chebyshev-Markov expansion (CME) [On83] and Arithmetical Fourier Transform (AFT) [Che10]-are indeed different looks of one entity, by the plausible missing link-Romanoff-Wintner theory (RWT). RWT generalizes both approaches, C...

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Published inHardy-Ramanujan Journal Vol. 43 - Special...; pp. 56 - 68
Main Authors Feng, J.-W, Kanemitsu, S, Kuzumaki, T
Format Journal Article
LanguageEnglish
Japanese
Published Centre pour la Communication Scientifique Directe (CCSD) 06.05.2021
Hardy-Ramanujan Society
Subjects
2010 Mathematics Subject Classification. 42A16, 11A25, 01A55
[MATH]Mathematics [math]
Arithmetical Fourier Transform
Chebyshev-Markov expansion
Hilbert space
Mathematics
Möbius inversion
Chebyshev-Markov expansion
Arithmetical Fourier Transform
Hilbert space
Möbius inversion
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Summary:In this survey-type paper we show that the seemingly unrelated two fields-Chebyshev-Markov expansion (CME) [On83] and Arithmetical Fourier Transform (AFT) [Che10]-are indeed different looks of one entity, by the plausible missing link-Romanoff-Wintner theory (RWT). RWT generalizes both approaches, CME and AFR, and was developed in [Wi44] and [Ro51a], [Ro51b] which were written independently. These two lost researches are very closely related and effective for producing new number-theoretic identities. Cf. [CKT09] for fragmental restoration of them.
ISSN:2804-7370
2804-7370
DOI:10.46298/hrj.2021.7426