1. In a paper “On the Fourier Constants of a Function,” published in the ‘Proceedings’ of this Society, I showed how certain theorems, previously given by myself, might be employed to obtain formulæ for the sums of certain series involving the Fourier constants of a function. In the case in which the function is bounded, it was only proved that the formulæ hold whenever the series converge, or, more generally, when the summation is performed in the Cesàro manner (index unity). In a more recent paper, also published in the ‘Proceedings’ of this Society, “On a Mode of Generating Fourier Series,” I showed incidentally that the formulæ are still applicable when the function has every power of index less than 1 +
p
summable, provided only the index
q
, which occurs in the formulæ in question, is greater than 1/(1 +
p
). Here again the theorem, as stated, contained the restriction that the summation was to be made in the Cesàro way. The main object of the papers in question was, in fact, to explain how certain methods might be employed, and these methods were in themselves inadequate for the purpose of removing the restriction. The theorems used involve the general theory of the integration of the Fourier series of a function term-by-term, when multiplied by another function. The coefficient
n
-q
in the series Ʃ
n
= 1
n
-q
a
n
and Ʃ
n
= 1
n
-q
b
n
considered, is, however, itself the typical Fourier constant both of an odd and an even function, which may be expected to possess special properties bearing on the matter in hand. A careful scrutiny of these properties has accordingly enabled me to take the step of removing the restriction above explained. The former of the main results, above stated, may be otherwise expressed by saying that, though the Fourier series of a bounded function need not converge, even if the function be continuous, it, and its allied series, will be made to converge, by dividing its coefficients by any power, however small, of the index denoting their respective places in the series. This affords a convenient necessary test that a given Fourier series is the Fourier series of a bounded function. In the same way we have a corresponding necessary condition that a Fourier series should have a function whose (l+
p
)th power is summable for associated function.
On the convergence of certain series involving the fourier constants of a function
