2. QUANTITATIV E RELATION S 3
D*(N; 1~) = 0 , while for 6(N; a) th e constan t i s chosen s o that /
T
S(N; a) da =
0. Similarl y
D(N;a,0)=6(N;(3)-6(N;a)
whenever a an d (3 ar e chose n s o as no t t o coincid e with an y u n. Henc e
D(N) = sup5(JV; (3) - in f 6(N\ a).
Prom thi s i t i s evident tha t
(12) S(N) D(N) 26(N).
2. Quantitativ e relations . W e now consider the possibility o f constructin g
quantitative relation s betwee n th e assertion s (a)-(d ) o f Weyl's criterion .
Suppose tha t F i s of bounded variatio n o n T, and tha t F i s continuous a t th e
points u
n
. The n
N
£ K ) = f F(a)dZ(N;a).
n = l
^
T
On subtractin g N Jj F(a) da fro m bot h sides , we deduce tha t
V F(u n) -N f F(a) da= f F(a) dD(N; a).
n = l J t Jj
Put
A(N;a) = D(N;a) - \ supD(N;/?) -\hdD(N;l3).
Thus supa A(7V; a) = ±D{N) an d infa A(JV ; a) = -%D(N). Bu t A(AT; a) differ s
from D(N\a) b y a constant , s o the integra l o n the righ t abov e i s
= [ F(a)dA{N-a).
JT
By integratin g b y part s i t follow s tha t thi s i s
= - [ A(N;a)dF{a).
JT
This integra l ha s absolut e valu e no t exceedin g
~D(N) J 1 \dF(a)\ = \D(N)Vzr T(F).
That is ,
N
(13)
J2F(un)~N J F(a)da
n = l
^
T
^D(N)VM
T
(F).
Although th e clas s o f Riemann-integrabl e function s include s function s no t o f
bounded variation , fo r practica l purpose s th e inequalit y abov e meet s ou r need s
in passin g fro m (d ) t o (c ) i n Weyl' s criterion .
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