Generalizing the Lomb-Scargle periodogram—the nonsinusoidal case

After finishing the paper generalizing the Lomb-Scargle periodogram to the case of quadrature data having a decaying sinusoidal signal [4]. I sent a copy of that paper to Jeffrey Scargle thinking that he would enjoy seeing how his periodogram could be generalized. He immediately asked whether or not the periodogram could be generalized to periodic but nonsinusoidal functions. The answer to this is of course yes, but for me it was not a particularly interesting case simply because I work in an NMR lab and in NMR the signals are so nearly sinusoidal that issues concerning nonsinusoidal oscillations never come up. However, in Astrophysics and a host of other applications the issue does come up. So I responded to his email message with a short note that explained how the Lomb-Scargle periodogram could be generalized to the nonsinusoidal but periodic case and then mapped out how those generalizations would go and what the sufficient statistics derived using Bayesian probability theory would be. In this paper, I give a refined version of that calculation and show that the Lomb-Scargle periodogram can be generalized in a way that covers both the sinusoidal and nonsinusoidal cases as well as a host of cases.