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, 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.