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 u1, u2, …, un
, 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 kth.
(iii) The sum of squared coefficients which measures the mean square error of yi
, is to be a minimum for each value of i.