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Abstract. We develop a general framework for the frequency analysis of irregularlysampled time series. It is based on the Lomb–Scargle periodogram, butextended to algebraic operators accounting for the presence of a polynomialtrend in the model for the data, in addition to a periodic component and abackground noise. Special care is devoted to the correlation between thetrend and the periodic component. This new periodogram is then cast into theWelch overlapping segment averaging (WOSA) method in order to reduce itsvariance. We also design a test of significance for the WOSA periodogram,against the background noise. The model for the background noise is astationary Gaussian continuous autoregressive-moving-average (CARMA) process,more general than the classical Gaussian white or red noise processes. CARMAparameters are estimated following a Bayesian framework. We providealgorithms that compute the confidence levels for the WOSA periodogram andfully take into account the uncertainty in the CARMA noise parameters.Alternatively, a theory using point estimates of CARMA parameters providesanalytical confidence levels for the WOSA periodogram, which are moreaccurate than Markov chain Monte Carlo (MCMC) confidence levels and, belowsome threshold for the number of data points, less costly in computing time.We then estimate the amplitude of the periodic component with least-squaresmethods, and derive an approximate proportionality between the squaredamplitude and the periodogram. This proportionality leads to a new extensionfor the periodogram: the weighted WOSA periodogram, which we recommend formost frequency analyses with irregularly sampled data. The estimated signalamplitude also permits filtering in a frequency band. Our resultsgeneraliseand unify methods developed in the fields of geosciences, engineering,astronomy and astrophysics. They also constitute the starting point for anextension to the continuous wavelet transform developed in a companionarticle ( Lenoir and Crucifix ,  2018 ) . All the methods presented in this paper areavailable to the reader in the Python package WAVEPAL.

A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 1: Frequency analysis