IV.—On Least Squares and Linear Combination of Observations

In a series of papers W. F. Sheppard (1912, 1914) has considered the approximate representation of equidistant, equally weighted, and uncorrelated observations under the following assumptions:– (i) The data being u 1 , u 2 , …, u n , the representation is to be given by linear combinations (ii) The linear combinations are to be such as would reproduce any set of values that were already values of a polynomial of degree not higher than the k th. (iii) The sum of squared coefficients which measures the mean square error of y i , is to be a minimum for each value of i .