Some years ago also I have tried to find the Fourier trasform for the mean values of the Time Equation and of the Sun's Declination for each day of the year; I remember to have sent a long message to this list in the February 1998.
Subsequently I have calculated the exact values of the quantities (at 11.00 UT) in each day for the years 2000 - 2047 and, from the mean values obtained from them, I have calculate the coefficients of the Fourier series. The values of the coefficients were also published in a book and in an article (about 1999-2000) These values are slightly different (but very similar) to those in John Davis' Glossary. After I have found the maximum errors in such a way : I have calculated, with these series, the Declination and the Time Equation on each day of the 48 years (2000-2047) and after I have compared the values obtained with the exact mean values calculated with the complete Meeus method. The maximum errors found in this experimental way result = ± 15 ' for the declination and ± 18 sec for the Equation of the time. For this reason I think that the error of 0.0006 rads ( = 2.1 ') is very optimistic. The Fourier transform that I have gotten ( with the quantities in degrees) is the following: D=0.3831+23.2600cos(w-169.785)+0.3551cos(2w-175.6692)+0.1342cos(3w-148.0399) + +0.0326cos(4w+2.9524) w=(N)*360/365.2421897 = 0.98564736*(N) N=1 for 1/1 ; 32 for 2/1, etc. In radians and with sin and cos : D=0.006686 - 0.399528 cosw + 0.071997 sinw - 0.006180 cos2w + 0.000468 sin2w -0.001987 cos3w + 0.001240 sin3w + 0.000568 cos4w - 0.000029 sin4w Some examples for Sun's Declination : March 21 - N=80 Davis =18.48 ' My result = 22.20 ' Mean value of the exact values (calculated on 3/21 in the years 2000-2047) = 22.44 ' 3/ 31 - N=90 Davis= 4d 13.15 ' My result = 4d 18.92 ' Exact mean value= 4d 17.48 ' 6/1 - N=152 Davis = 22d 4.43 ' My result = 22d 6.08 ' Exact mean value = 22d 6.0 ' For the calculation of the exact values I have used my program SUNEPH_0103 , distributed as bonus with "The Compendium", and that can be downloaded from our site : http://www.gnomonicaitaliana.it/index.php?area=15&article=57 To calculate the declination in hours different from 11.00 UT, it is necessary to take N = Number of the day + (Hour_UT - 11) / 24 and to use the decimal number so obtained. For instance wanting the values for 5 pm UT (noon at New York) we have to take N= Number of the day +0.25 Gianni Ferrari P.S. If someone is interested I can send the Fourier transform for of the Equation of the Time -
