Type: Article
Publication Date: 1911-01-01
Citations: 54
DOI: https://doi.org/10.1112/plms/s2-9.1.449
In a companion paper * presented to the Society, I have discussed the circumstances under which one of the factors of the integrand of a definite integral may be replaced by a series, which is then multiplied term-by-term by the other factor of the integrand, and the new series so> obtained integrated term-by-term.It was there pointed out that the series substituted for the factor in question need not have that factor for sum, and it need not, indeed, even converge.The more novel of the theorems obtained related, in fact, precisely to this case.Now, the series, of Fourier can only be shown to converge, even in general, in a very restricted class of cases.On the other hand, what may conveniently be called the integrated Fourier series, obtained from the original series by replacing each term by its simplest indefinite integral, necessarily converges uniformly to the integral of the function associated with the original series.In accordance therefore with one of the theorems of the paper cited, we are at liberty to substitute for any function, which is summable in Lebesgue's manner, constituting one of the factors of the integrand of an integral, its corresponding Fourier series, whether this last converges or not, provided only that the remaining factor of the integrand is a function of bounded variation.This result is, however, the only one that follows immediately from the theorems of the paper cited.There are, none the less, a number of interesting and important cases, distinct from this one, in which the mode of procedure in question may be adopted.It is the main object of the present paper to set these forth in order.As far as I know, this has not been attempted in any existing textbook or memoir.In Hobson's Treatise the subject is not directly touched on, and earlier writers were necessarily ignorant of the considerations from which these results follow.
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