An Application of Fourier's Series to Probability.
David F. Barrow · Annals of Mathematics · 1917
Many persons have been interested in compounding frequencies. They all seem to aim at asymptotic expressions which represent approximately the functions obtained by compounding a large number of frequencies. The most general theorem of this sort is to the effect that if a great many frequencies be compounded the result approaches a Gaussian error function.* The generality and beauty of this theorem, and the fact that in most applications a great many frequencies enter, seem to have led writers to give little attention to those cases where only a few frequencies enter. Furthermore in those cases where exact formulas for compounding a few frequencies have been given, great difficulty is found in obtaining numerical results. Even Laplace confesses that in one case such a difficulty arrested him for a long time before he found a suitable approximation. This article gives a simple method of compounding frequency functions when they can be represented by Fourier's series. It has the advantages of being applicable over a wide range of problems and of yielding formulas which represent exactly the frequencies and from which numerical results can be obtained with relative ease. Owing to the existence of asymptotic formulas, it is most useful where a few frequencies are to be compounded, that is where the asymptotic solution is not very exact. In section 1 the general method is set forth. In section 2 a particular problem is solved. The solution looks so unlike the asymptotic solution that it seems of interest to compare the two directly to get an upper limit on the discrepancy. This is done in sections 4 and 5, and also a way is pointed out for comparing the exact and asymptotic solutions in various other cases.