Ramanujan sums analysis of long-period sequences and 1/f noiseM. Planat1, M. Minarovjech2 and M. Saniga2
1 Institut FEMTO-ST, CNRS - 32 Avenue de l'Observatoire, F-25044 Besançon, France, EU
2 Astronomical Institute, Slovak Academy of Sciences - SK-05960 Tatranská Lomnica, Slovak Republic, EU
received 11 December 2008; accepted in final form 30 January 2009; published February 2009
published online 4 March 2009
Ramanujan sums are exponential sums with exponent defined over the irreducible fractions. Until now, they have been used to provide converging expansions to some arithmetical functions appearing in the context of number theory. In this paper, we provide an application of Ramanujan sum expansions to periodic, quasi-periodic and complex time series, as a vital alternative to the Fourier transform. The Ramanujan-Fourier spectrum of the Dow Jones index over 13 years and of the coronal index of solar activity over 69 years are taken as illustrative examples. Distinct long periods may be discriminated in place of the spectra of the Fourier transform.
02.10.De - Algebraic structures and number theory.
05.45.Tp - Time series analysis.
89.20.-a - Interdisciplinary applications of physics.
© EPLA 2009