Methods of local approximations are widely used for smoothing the signals of different nature (see classic textbook by Whittaker & Robinson (1928) and e.g. a recent monograph by Hardle (1990) and papers by Foster (1996a,b). We point our attention towards a particular class of smoothing functions and discuss their properties by approximating harmonic signals and the "white noise".
The method of "running parabolae" for smoothing signals with both
equidistantly and not equidistantly distributed in time signals was
proposed in our Paper I (Andronov 1990) and was applied to light curves
of stars of different types, e.g. HQ And (Andronov et al. 1992a), MV Lyr
(Andronov et al. 1992b), TT Ari (Tremko et al. 1996), UV Aur, TX CVn,
V1329 Cyg (Chinarova et al. 1994). The advantage of this method as
compared with, e.g. smoothing by a running mean is a smooth
approximating curve, which has continuous first derivative and a better
amplitude-frequency dependence. This allows application of the method
to aperiodic and cyclical processes and, particularly, to determine
extrema. In this work we analyse properties of the smoothing function in
more detail, comparing 4 modifications, namely:
1) "um", unweighted mean, usually referred to as a "running mean"
or "moving average";
2) "wm", weighted mean, with weights (5);
3) "up", unweighted parabolae with constant weight;
4) "wp", weighted parabolae with weights (5), called "running parabolae"
in Paper I.
Hereafter in the text we will use two-letter abbreviations, whereas in figures - the numerical ones.