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

18-53- 18-54]

1 FOURIER —BESSEL SERIES

613

Now this upper bound for I xSSn (x I R) I does not exceed
arv# (x dt + 8 c11 3A,2 o (n ))
4k_6 X _ — __ - ___
A 2f 0 {t-f(t)- - f(() x dt +V2 '
and, since x-,f(x) is bounded (because it is continuous), this can be made
arbitrarily small by a choice of n which is independent of x.
Consequently x*St (x 1 R) tends to zero uniformly as n — o.
Now it has already been shewn (~ 18'22) that
xa X tv+lTn(t,x)dt
0o
is uniformly convergent in (, 1 - A), and so, since uniformity of convergence
involves uniformity of summability,
X f(x) ft Tn (t, X R) dt
tends uniformly to x-f(x) in (0, b - A).
Hence, since x- Sn (x R) tends to zero uniformly,
X2 t tf () n(t, R) dt
/o
tends uniformly to xi-f (x) J tv+ (t, x R) dt, i.e. to xf (x) in (0, b -A).
It has therefore been proved that
00
E axJi (j mc)
m=l
is uniformly summable (R) in (0, b- A) with sum xf (x), provided that
fotlf(t) dt
exists and is absolutely convergent, and that t-f(t) is continuous in (0, b).
18'54. Methods of 'summing' Fourier-Bessel series..
We shall now investigate various methods of summing the Fourier-Bessel
series *
S amJ, (jo,,,x)
m=o
on the hypotheses (i) that the limits f(x + 0) exist, (ii) that
f tf(t)dt
J o
exists and is absolutely convergent, and (iii) that the series is summable (R).
It conduces to brevity to write fm (x) in place of axtJ^, (x), so that fm ()
tends uniformly to zero (~ 18'27) as m-moo when x lies in (0, 1).
* The factor x] is inserted mierely in order that the discussion may cover the investigation of
uniformity of summability near the origin.