On least squares discrete Fourier analysis of unequally spaced data

Tom 47 / 2020

Waldemar Popiński Applicationes Mathematicae 47 (2020), 207-224 MSC: 65T40, 65T50. DOI: 10.4064/am2399-6-2020 Opublikowany online: 9 November 2020

Streszczenie

The problem of discrete Fourier analysis of observations at non-equidistant times using the standard set of complex harmonics $\exp (i2\pi kt)$, $t\in \mathbb {R}$, $k=0,\pm 1,\pm 2,\ldots ,$ and the least squares method is studied. The observation model $y_j = f(t_j) + \eta _j$, $j=1,\ldots ,n$, is considered for $f\in L^2[0,1]$, where $t_j\in [(j-1)/n,j/n)$, and $\eta _j$ are correlated complex valued random variables with $E_\eta \eta _j=0$ and $ E_\eta |\eta _j|^2=\sigma _\eta ^2 \lt \infty $. Uniqueness and finite sample properties of the observed function Fourier coefficient estimators $\hat c_k$, $k=0,\pm 1,\ldots ,\pm m$, where $m \lt n/(8\pi )$, obtained by the least squares method, as well as of the corresponding orthogonal projection estimator $\hat f_N(t)=\sum _{k=-m}^m\hat c_k\exp (i2\pi kt)$, where $N=2m+1$, are examined and compared with those of the standard Discrete Fourier Transform.

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