On the Theory of the Application of Expansions to Definite Integrals
William H. L. Young · Proceedings of the London Mathematical Society · 1911
In the application of expansions to definite integrals, either the whole integrand or one of its factors is replaced by an infinite series.The new infinite series, now constituted by the integrand, is then integrated term-by-term.The special case in which the integrand consists of a single function will not be discussed here ; the problem of determining whether the substitution is in that case allowable is merely the general problem of the term-by-term integration of infinite series.In the more general case, the series which we substitute for one of the factors may converge everywhere to that factor as sum, or it may do so except at a set of content zero, or it may not have that function as sum, or even converge other than at exceptional points.We have an example of the last-named case when we substitute for one of the factors its series of Fourier; if the factor in question is a function of a general character, its Fourier series will not converge at all.On the other hand, the series got by integrating the Fourier series term-by-term always converges ; it converges, in fact, uniformly to one of the integrals of the function associated with the original series.The example given by the application of the Fourier expansion suggests the general problem.Let the integraud of the integral to be considered be f{x)g(x).If fix) can be expanded in a converging series, may we integrate this series when multiplied term-by-term by gix) ?If we cannot conveniently replace f(x) by such a series, expand its integral F(x) and differentiate the series so obtained term-by-term.Can we then integrate term-by-term the series got by multiplying termby-term this last series by g(x), and, if so, shall we in this way obtain \ fix) gix)dx ?If Fix) cannot be conveniently expanded, and its integral can, take the series representing this latter integral and differentiate it twice.Can the new series, when multiplied term-by-term by gix), be integrated term-by-term, and is the sum in this case j fix) gix)dx ?* For brevity we have not usually distinguished between a sequence and a succession which converges except at a set of content zero.For the same reason I have not specially called attention to the fact that the functions which occur in the theorems or in the processes may, under certain circumstances, only exist at a set complementary to a set of content zero.| We say that a sequence converges boundedly if s n (x) is a bounded function of the ensemble (z, n), so that there are no points of non-uniform convergence with infinite measure; in other words, that the peak and chasm functions are finite.The boundedness must, of course, be without exception, even when non-convergence is allowed at a set of points of content zero, in accordance with the previous footnote.