  Many astrophysical phenomena are time-varying, in the sense that their
brightness change over time. In the case of periodic stars, previous approaches
assumed that changes in period, amplitude, and phase are well described by
either parametric or piecewise-constant functions. With this paper, we
introduce a new mathematical model for the description of the so-called
modulated light curves, as found in periodic variable stars that exhibit
smoothly time-varying parameters such as amplitude, frequency, and/or phase.
Our model accounts for a smoothly time-varying trend, and a harmonic sum with
smoothly time-varying weights. In this sense, our approach is flexible because
it avoids restrictive assumptions (parametric or piecewise-constant) about the
functional form of trend and amplitudes. We apply our methodology to the light
curve of a pulsating RR Lyrae star characterised by the Blazhko effect. To
estimate the time-varying parameters of our model, we develop a semi-parametric
method for unequally spaced time series. The estimation of our time-varying
curves translates into the estimation of time-invariant parameters that can be
performed by ordinary least-squares, with the following two advantages:
modeling and forecasting can be implemented in a parametric fashion, and we are
able to cope with missing observations. To detect serial correlation in the
residuals of our fitted model, we derive the mathematical definition of the
spectral density for unequally spaced time series. The proposed method is
designed to estimate smoothly time-varying trend and amplitudes, as well as the
spectral density function of the errors. We provide simulation results and
applications to real data.
