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54
Proof. This is almost trivial if b> N
N k k+1
V. The Waring Problem 2/3 , for, since the derivative
of | eàxn )| is bounded by 2πN ,
n 1

N N
a
k eà k 2πN
a k+1
b b
eàxn ) ≤ n ) + x −

n 1 n 1
N 1+o(1) 2πN 3/2 N 1+o(1) 2πN
≤ + ≤ + ,
1 1 1/4
b 2 k−1 b b 2 k−1 b
by C, which gives the result, since j 0 automatically. Assume
therefore that b ≤ N 2/3 , and note the following two simple facts (A)
and (B). For details see [K. Knopp, Theory and Application of Infinite
Series, Blackie & Sons, Glasgow, 1946.] and [G. P´ olya und G. Szeg¨ o,
Aufgaben und Lehrs¨ atze aus der Analysis, Dover Publications, New
York 1945, Vol. 1, Part II, p. 37]. Q.E.D.
m
(A) If M is the maximum of the moduli of the partial sums a n ,
n 1

V the total variation of fàt) in 0 ≤ t ≤ N, and M the maximum
of the modulus of fàt) in 0 ≤ t ≤ N, then

N

a n f (n) ≤ M(V + M ).

n 1
(B) If V is the total variation of fàt) in 0 ≤ t ≤ N, then

N N

f (n) − fàt)dt ≤ V.

0
n 1
1 b a k
Now write α eà n ) and
b n 1 b
N

k
eàxn ) S 1 + αS 2 , à 3)
n 1
where
N
a k a k
S 1 e n − α e x − n ,
b b
n 1
N
a
S 2 e x − n k .
b
n 1