A treatise on the theory of Bessel functions, by G. N. Watson.

18-26, 18-27]

FOURIER-BESSEL SERIES

595

Hence we can make
f (x) - x-v f (x) tV+1 Tn (t, x) dt
arbitrarily small for all values of x in (a + A, 1) by a choice of n which is
independent of x; and this establishes the uniformity of the convergence of
tf(t)Tn(t, x)dt
to the sum f(x) in (a + A, 1) in the postulated circumstances.
18'27. The order of magnitude of the terms in the Fourier-Bessel series.
It is easy to prove that, if tf (t) has limited total fluctuation in (a, b),
where (a, b) is any part (or the whole) of the interval (0, 1), then
ftf(t) Jv(\t) dt =O(
as -oo.
From this theorem we at once obtain Sheppard's result* that
2 v+ (jm ) Jt mf (t) Jv (jt) dt =
when 0 < x 1; this equation, of course, has a well-known parallel in the
theory of Fourier series.
We first observe that, as a consequence of the asymptotic expansion of ~ 7'21,
f| ttJ(t)dt < c,
where c is a constant, independent of t when t lies in the interval (0, oo).
Now write ttf(t) = f (t) - r, (t), where AH (t) and 2 (t) are monotonic
in (a, b); and then a number 5 exists such that
| | (t) t9 Jv (Xt) dt = (a)l tO J (Xt) dt + i1 (b) ft Jv (Xt) dt
<2c |1 (a) |+|1(b) i} X= o=0(X- ).
A similar result holds for 2 (t), and hence the theorem stated is evident.
If it is known merely that
f tf(t) dt
exists and is absolutely convergent, then all that can be proved is the
theorem that
fjtf(t) (t J (t) dt = o (1/X).
* Quarterly Journal, xxII. (1889), p. 247.
38-2