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

18-34]

DINI SERIES

603'

For any given positive value of s, it follows from ~ 18-21 that this is a
bounded function of t in the interval (8, 1). When 3 < t - 1 - 8, it is 0 (1/Dn).
And when t =1, it has the limit 1 when n-oo.
It follows that
Xo (1) + 2 b,J, (X,)-) =f( -0)f tV+' t-vf(t)-f(1 - 0)} Tn (t, 1; H) dt.
Since t-"f(t) -f( - 0) has limited total fluctuation in (a, 1) we may
write it in the form X% (t)- X (t), where Xi (t) and X, (t) are bounded positive
decreasing functions of t such that
XI( -0)= X2( - 0)= 0.
Hence, given an arbitrary positive number e, we can choose a positive
number 8, not exceeding 1 - a, such that
o <x,(t)<, 06< X2(0<
whenever 1 - 8 t < 1.
We then have!
Jtv+i {t-vf(t)-f(1 - 0)} Tn (t, 1; H) dt
o
= | -t + {f(t-)(-f(l - O)} T (t, 1; H) dt
+ f t^+1 (t) Tn (t, 1; H) dt - t +, X2 (t) T (t, 1; H) dt.
By arguments similar to those used in ~ 18'24, the first integral on the
right is o (1) as n- oo; and neither the second nor the third exceeds
2e lim tL Tn (t, 1; H) dt
in absolute value (cf. ~ 1824), and this expression is arbitrarily small.
It follows that
lim t+1 {t-"f(t) -f(1 - 0)} T (t, 1; H) dt = 0,
w —oo d 0
and so we have proved that, in the circumstances postulated at the beginning
of this section,
o (1)+ z bm J, (xn)
m=l
converges to the sum f(1 - 0).
This discrepancy between the behaviours of Dini series and of FourierBessel series (~ 18'26) is somewhat remarkable.