On the series of Legendre

In recent communications to the Society, I have confined myself largely to the Theory of Fourier Series, partly because much seemed to me still to require doing in this subject, partly because I believed its thorough investigation to be the natural preparation for the study of other series of normal functions. It has, indeed, been known for some time that the behaviour of, for instance, series of Sturm-Liouville functions exactly corresponds to that of Fourier series. The introduction that I have recently made into Analysis of what I have calledrestricted Fourier series enables us to notably extend the range of such analogies. I propose in the present communication to illustrate this remark with reference to series of Legendre coefficients. Whereas Fourier series may be said to be “naturally unrestricted,” in virtue of the fact that the convergence of the integrated series to an integral necessarily involves the tendency towards zero of its own general term, so that the consideration of the more general type of series does not at once suggest itself, Legendre series may be said to come into being “restricted,” even when the coefficients are expressible in what may be called the Fourier form by means of integrals involving Legendre’s coefficients. In other words, such series correspond precisely to restricted Fourier series, instead of to ordinary Fourier series like the analogous series of Sturm-Liouville functions.