ANALYSIS OF LEAST SQUARES SMOOTHING OPERATORS IN THE FREQUENCY DOMAIN*

The process of data smoothing by least squares operators with real and symmetric coefficients is shown to be equivalent to linear digital filtering. Study of the frequency response functions of the least squares operators shows that these operators are essentially low‐pass filters. The frequency response functions are characterized by the presence of a main lobe centered about the zero frequency together with a number of minor lobes of smaller amplitudes at higher frequencies. The predominant band of frequencies which is passed by a least squares operator occurs within the main lobe of the frequency response function. The filtering characteristics of a least squares operator are dependent on the degree of the polynomial and the number of data points. Generally speaking, the higher the degree of the polynomial the broader the pass band of the filter will be. But the effect of the number of data points on the pass band is exactly opposite.

Several figures showing the plots of the half‐power frequency, the quarter‐power frequency, and the half‐width of the main lobe for a large number of least squares operators with different combinations of the degree of the polynomial and the number of data points are presented in this paper. These figures will be helpful to those people who wish to make a selection of least squares operators suitable for their purposes.