4. Evenly sampled data and limits for continuous functions

The mentioned above equations are valid without restrictions on the distribution of the times of observations. However, evenly spaced signals are also often used in astronomy, and they allow to use the same matrices inside the interval, except the edges. For large number of evenly sampled observations, one may replace sums by integrals:

Corresponding parameters may be computed as integrals

Particularly, an expectation of the rms deviation of the parameter X is

where V[X] is a value of the integral characterizing variance of X, and is an "unit weight" error (cf. Whittaker & Robinson 1928). Hereafter one may omit a constant n/(z₂-z₁), when replacing sums by integrals, while not specially mentioned. For example, Eq. (9) may be rewritten as

For an interval (-1, 1) covered by the observations, S_s=2/(s+1) for even s and 0 else. For constant weights p(z)=1 (case "up") the matrices are the following:

For the coefficients one may obtain

For simple running mean ("um") h[C₀,z]=1/2, V=1/2, and other coefficients by definition. One may note that coefficients C₁ and C₂ for "unweighted" polynomial fits are equal to and (in dimensionless argument units z). However, with changing interval, the observational points are added to and removed from the set, thus such an approximation is valid only for a fixed set of data (t_k,x_k). For continuous functions, one may not choose an interval of t₀ with a fixed set, and derivatives are computed using Eq. (31).

For the weights p(z)=(1-z²)²,

For "weighted mean" ("wm"):

For "weighted parabolae" ("wp"):

To determine h[C₀₁,z] and h[C₀₂,z], one would use Eqs. (38, 39). However, for continuous polynomial fits and all weight functions satisfying the condition the matrices are equal to zero, as well as For "wp",

One may note that more complicated character of the evaluation of the derivatives as compared with coefficients C₁ and C₂ leads to much larger values of corresponding parameter V, e.g. V[C₀₁]/V[C₁₀]=525/140.

  
Figure 3: Projective functions h[C₀,z] for 4 model "symmetric" polynomial fits

First derivative of the smoothing function in the case of continuous signal x(t) is equal to

For "symmetric" fits one may obtain

One may note that derivatives for the "unweighted" fits are strongly dependent on particular values of the signal at the borders, making impossible the form (50) without using the Dirac's functions. In a case one may introduce corresponding functions and to obtain V=15/7 for "wm" and V=525/22 for "wp".

Expressions for "asymmetric" fits are much more complicated. For an extreme case for "unweighted" fits, and . Corresponding functions h are equal to 0, if z<0 and z>1. From 0 to 1 they are the following: