Adaptive Fourier series and the analysis of periodicities in time series data (Q2740044)

The author considers an approximation of a function (time series) \(x(t)\simeq\sum_{k=-m}^m x_k(t)\), where \(x_k(t)\) is a function from \(B_{k,n}\), a linear span of NEWLINE\[NEWLINEf_{k,n,j}(x)=\exp(i\pi 2^{-1}[2kt/\pi+t_{j,n}]),\;j=1,\dots,n,NEWLINE\]NEWLINE where \(t_{j,n}=2/\pi-(j-1)/n+1.\) All functions in \(B_{k,n}\) are periodic with the period \(2\pi/k\). The idea of such an approximation is to derive a Fourier-like analysis for non-harmonic periodicities. Projective procedures of the approximation for functions and time-series data are described, and theorems of their uniqueness are demonstrated. An ANOVA technique based on such approximations is considered analogous to Fourier-series ANOVA. It is applied to electroencephalographic sleep recordings and to body weight measurements.