On the Convergence of Fourier‐Bessel Series

where, in the notation of Watson's book on Bessel functions!, * e expression w !r denotes the Bessel function of order v of the first kind, and j m is its m-th positive zero in ascending order of magnitude. The index v is fixed in (1.1) and we suppose, with Watson f, that (1.3) v+i^Q. It is convenient to stipulate further, unless the contrary is explicitly stated, that we have (1.4) x*f(x) = 0 when x = 0 and when x= 1, in view of the fact that the functions (1.1) vanish at the end points 0 and 1. The problem of the convergence of our series in the partial interval e 0, may be regarded as settled by the memoir of