Integration of Fourier-Legendre series

Abstract

It is well known than an ordinary (trigonometric) Fourier series “can” be integrated term by term; that is, whether the Fourier series of an integrable function \(f\) is convergent or not, the series obtained by term by term integration of it is convergent to the corresponding integral of \(f\). There is also a more general theorem in which \(f\) and its Fourier series are first multiplied by a function of bounded variation before integrating. This paper aims to obtain similar theorems for the Fourier-Legendre series discussed by Love and Hunter in Proc. London Math. Soc (3) 64 (1992) 579-601 and by Love in that journal (3) 69 (1994) 629-672. Most of the paper is occupied by lemmas which lead to establishing the dominated convergence of the partial sums of the Fourier-Legendre series under certain conditions. For several reasons one of these conditions is that the interval of integration must be a closed subinterval of \((-1,1)\), the interval on which the Legendre functions \(P_\nu^\mu (x)\) are defined; further, I have only had success with integration over subintervals of \([- \frac{1}{2} \sqrt{3}, \frac{1}{2}\sqrt{3}]\).