Analytic expressions in a general form are derived for the expectation of functions related to the smoothing the astronomical signals using local approximations with additional weights. No restrictions are made on distribution of the times of the signal. Applications are made for polynomial fits of orders 0 and 2 and weights $p(z)=1$ and $(1-z^2)^2$. These variable weights ensure that the smoothing function will be both continuous and differentiable, which is important for determining the extrema and the shape of the variations. Special attention is paid to evenly spaced time series data, which if their number is large enough will allow one to obtain analytic expressions for the main functions.