The double-averages (20) of the columns in the triangular array (19) of products of data—these averages obtained, indeed, without reference to any particular period—are, nevertheless, in a well-defined sense, graduated values of the data for all periods present, as seen from (23) and (24).
These averages in (19), moreover, may be used as approximations for the limit-averages in (16) and (17) when it is proposed to get the intensity (9) for a periodogram by a complete process of averaging, where each set of m consecutive rows in the array (12) is used, not merely each non-overlapping set.
The process here outlined, indeed, reverses the usual order of operations. Instead of selecting a trial period, applying the proper harmonic multipliers, and then averaging, we use first a general process of averaging in (19) and then for trial periods apply the harmonic multipliers in (16) or (17), where θ is given by (13), in testing for a period of k/1.
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