Abstract
Harmonic analyses for many terms are troublesome to perform by the ordinal scheme, and in this paper we showed that they are easily obtained by combining sets of simpler analyses.
When integers m and n are incommensurable, we represent m×n terms y0, y1, y2, ……, yp, ……, ymn-1 as follows.
where m is even.
(i) Calculate the Fourier coefficients of the j-th line of (I) using the analyses for m orders whose results are
(ii) Calculate the Fourier coefficients of the i-th column of (II) using the analyses for n orders whose result are
(iii) The Fourier coefficients of the entire series P (ν), and Q (ν) are shown by, and where
As examples we showed schematically the analyses for 36 (4×9), 60 (4×15), 72 (8×9), 120 (8×15) and 180 (5×36) orders, the results of which were given by the tables (IV. 36), (IV. 60), (IV. 72), (IV. 120), and (IV. 180) in the text. In each case mentioned above m is even, but in the case when both m and n are odd the similar scheme can be used.
