1. Series which converge expect at a set of content zero, or, using the expression very commonly adopted, series which converge usually, posses many of the properties which appertain to series which converge every-where. It becomes, therefore, of importance to device circumstances under which we can assert the consequence that a series converge in this manner. The subject has recently received considerable attention. so far as Fourier Series are concerned no result of even an approximately final character has been obtained. It may be supposed, indeed, that the result* of Jerosch and Weyl were at first so regarded, but, if we examine them closely in the light of the Riesz-Fischer theorems, which was known previously to the result of these authors, it becomes evident that they are merely
equivalent
to the statement that the Fourier Series of a function, whose square is summable, is changed into one which converges usually, if the typical coefficient a
n
and b
n
are divided by the sixth root of the integer
n
denoting their place in the series. Now it is difficult to believe that the question of the usual convergences of a Fourier Series can depended on the degree of the summability of the function with which it is associated and it is still more difficult to see how precisely the sixth root of
n
can have anything to do it. On the other hand Weyl's method, which itself marks an advance on that of Jerosch, does not obviously lend itself to any suitable modification which would secure a greater degree of generality in the result. The mistake is frequently made of confusing theoretical interest with pracitcal importance in the matter of a necessary and sufficient test. Tests which are only sufficient, but not necessary, are often much more convenient. Still more frequent it is convenient to work from first principles, and not to use any test at all. Instead of employing Weyl's necessary and sufficient condition that a series should converge usually, I have attacked the problem directly. The principles I have employed do not differ essentially from those already exposed in previous communication to this Society, but the generality and interest of the result obtained in the matter in hand seem to justify a further communication.
Hubert Maitland Turnbull, 1875-1955
